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MATHEMATICS STUDY GUIDE
Grade 7
A member of the FUTURELEARN group
Mathematics Study guide
1907-E-MAM-SG01
Í3’È-E-MAM-SG01NÎ
Grade 7
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DM Oost
Study Guide G07 ~ Mathematics
CONTENTS UNIT 1: NUMBER SYSTEMS ............................................................................................. 3 Exercise 1: Where do number systems come from? ..................................................... 3 Exercise 2: Natural numbers ......................................................................................... 4 Exercise 3: Counting numbers .................................................................................... 16 Exercise 4: Integers ..................................................................................................... 18 Exercise 5: Fractions (rational numbers) ..................................................................... 21 Exercise 6: Roots (irrational numbers) ........................................................................ 34 Exercise 7: Properties for calculations ........................................................................ 37 Exercise 8: Percentages.............................................................................................. 41 Exercise 9: Calculator work ......................................................................................... 44 Exercise 10: Mixed exercises ...................................................................................... 47 Bibliography................................................................................................................. 48 UNIT 2: EXPONENTS ....................................................................................................... 51 Exercise 1: Expanded exponential notation................................................................. 51 Exercise 2: Calculations with exponents ..................................................................... 53 Exercise 3: Prime factors............................................................................................. 55 Exercise 4: Calculator work ......................................................................................... 57 Exercise 5: Mixed exercises ........................................................................................ 59 Bibliography................................................................................................................. 59 UNIT 3: ALGEBRA ........................................................................................................... 62 Exercise 1: Variables ................................................................................................... 62 Exercise 2: Basic calculations ..................................................................................... 66 Exercise 3: Algebraic expressions ............................................................................... 70 Exercise 4: English to Maths ....................................................................................... 74 Exercise 5: Substitution ............................................................................................... 77 Exercise 6: Linear equations ....................................................................................... 79 Exercise 7: Inequality .................................................................................................. 85 Exercise 8: Basic word problems ................................................................................ 87 Exercise 9: Mixed exercises ........................................................................................ 90 Bibliography................................................................................................................. 91 UNIT 4: NUMBER PATTERNS AND RELATIONSHIPS................................................... 93 Exercise 1: Visual presentation of patterns ................................................................. 93 Exercise 2: Number patterns in a sequence ................................................................ 95 Exercise 3: Types of number patterns (sequences) .................................................... 98 Exercise 4: Determine rules for general terms .......................................................... 100 Exercise 5: Relationships: Input and output diagrams ............................................... 103 Exercise 6: Number coordinates (ordered pairs) ....................................................... 106 Exercise 7: Mixed exercises ...................................................................................... 111 Bibliography............................................................................................................... 113 © Impaq
Study Guide G07 ~ Mathematics
UNIT 5.1: GEOMETRY .................................................................................................... 116 Exercise 1: Measure and construct lines ................................................................... 116 Exercise 2: Measure and construct angles ................................................................ 122 Exercise 3: Different angles....................................................................................... 125 Exercise 4: Parallel lines ........................................................................................... 130 Exercise 5: Triangles ................................................................................................. 136 Exercise 6: Quadrilaterals ......................................................................................... 140 Exercise 7: Polygons ................................................................................................. 145 Exercise 8: Circles ..................................................................................................... 152 Exercise 9: Mixed exercises ...................................................................................... 153 Bibliography............................................................................................................... 155 UNIT 5.2: GEOMETRY: AREA, PERIMETER AND VOLUME........................................ 157 Exercise 1: Convert units........................................................................................... 157 Exercise 2: Rectangles .............................................................................................. 158 Exercise 3: Triangles ................................................................................................. 161 Exercise 4: Circles ..................................................................................................... 164 Exercise 5: Polygons ................................................................................................. 166 Exercise 6: Volume ................................................................................................... 169 Exercise 7: Surface area ........................................................................................... 171 Exercise 8: Mixed exercises ...................................................................................... 176 Exercise 9: Summary of formulas.............................................................................. 179 Bibliography............................................................................................................... 180 UNIT 6: TRANSFORMATION GEOMETRY .................................................................... 182 Exercise 1: Symmetry................................................................................................ 182 Exercise 2: Reflection ................................................................................................ 184 Exercise 3: Translation .............................................................................................. 186 Exercise 4: Rotation .................................................................................................. 189 Exercise 5: Enlargement and reduction ..................................................................... 192 Exercise 6: Mixed exercises ...................................................................................... 197 Bibliography............................................................................................................... 201 UNIT 7: RATIO AND RATE ............................................................................................. 203 Exercise 1: Ratios and equivalent fractions ............................................................... 203 Exercise 2: Divide into specific ratios ........................................................................ 204 Exercise 3: Rate ........................................................................................................ 207 Exercise 4: Increasing and decreasing values .......................................................... 208 Exercise 5: Distance, speed and time ....................................................................... 209 Exercise 6: Scale drawings ....................................................................................... 210 Exercise 7: Mixed exercises ...................................................................................... 211 Bibliography............................................................................................................... 211
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Study Guide G07 ~ Mathematics
UNIT 8: FINANCE ........................................................................................................... 213 Exercise 1: Calculations with money ......................................................................... 213 Exercise 2: Money and ratios .................................................................................... 213 Exercise 3: Percentages............................................................................................ 214 Exercise 4: Profit, loss and discount .......................................................................... 215 Exercise 5: VAT......................................................................................................... 216 Exercise 6: Interest .................................................................................................... 217 Exercise 7: Exchange rate......................................................................................... 218 Exercise 8: Mixed exercises ...................................................................................... 220 Bibliography............................................................................................................... 221 UNIT 9: STATISTICS ...................................................................................................... 224 Exercise 1: Collecting data ........................................................................................ 224 Exercise 2: Organisation and summary of data ......................................................... 229 Exercise 3: Presentation of data................................................................................ 232 Exercise 4: Analysis and calculation of data.............................................................. 240 Exercise 5: Interpretation and evaluation of data ...................................................... 243 Exercise 6: Mixed exercises ...................................................................................... 243 Exercise 7: Terminology ............................................................................................ 245 Bibliography............................................................................................................... 247 UNIT 10: PROBABILITY ................................................................................................. 252 Exercise 1: Terminology ............................................................................................ 252 Exercise 2: Relative frequency of an outcome .......................................................... 253 Exercise 3: The probability scale ............................................................................... 255 Exercise 4: Probability P(x) of outcome x .................................................................. 256 Exercise 5: Mixed exercises ...................................................................................... 257 Bibliography............................................................................................................... 258
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Study Guide G07 ~ Mathematics
Year plan: Abridged version Term T1 T1 T1 T2 T2
Content Number systems Exponents Geometry Number systems fractions Number patterns and relationships T2 Perimeter, area and volume T3 Algebra T3 Transformations T3 Ratio and rate T3 Finance T4 Statistics T4 Probability Revision November examination Paper covers the entire year’s work
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Unit in study guide 1 2 5.1 1 4 5.2 3 6 7 8 9 10
Study Guide G07 ~ Mathematics
Unit
1
CONTENTS Unit 1: Number systems ................................................................................................. 3 Exercise 1: Where do number systems come from? ..................................................... 3 Exercise 2: Natural numbers ......................................................................................... 4 Exercise 3: Counting numbers ..................................................................................... 16 Exercise 4: Integers ..................................................................................................... 18 Exercise 5: Fractions (rational numbers) ..................................................................... 21 Exercise 6: Roots (irrational numbers)......................................................................... 34 Exercise 7: Properties for calculations ......................................................................... 37 Exercise 8: Percentages .............................................................................................. 41 Exercise 9: Calculator work ......................................................................................... 44 Exercise 10: Mixed exercises ...................................................................................... 47 Bibliography ................................................................................................................. 48
Simplify means...
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Study Guide G07 ~ Mathematics
Unit
Level of difficulty 1 Basic knowledge 25%
*
Level of difficulty 2 Routine procedures 45%
**
Level of difficulty 3 Complex procedures 20%
***
Level of difficulty 4 Problem-solving 10%
****
Level of difficulty 5
**** *
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Assessment analysis • Use mathematical facts and vocabulary • Use the correct formulas • Predict answers and round off values • Theoretical knowledge • Complete known procedures • Apply knowledge and facts with more than one step • Come to conclusions from given information • Basic calculations as learned from examples and exercises • Complex calculations and high order arguments • Euclidean Geometry • No stipulated path to follow • Similarities and differences between presentations • Need conceptual and holistic approaches to problems • Unseen and not-routine problems. • Problems in different sections • Still in curriculum and study guide • Mostly aimed at practical and everyday situations. • High order of thinking Advanced Often regarded as acceleration of the curriculum. Not included in tests and exam papers.
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Study Guide G07 ~ Mathematics
Unit
1
Unit 1: Number systems Content in this unit: • Where do number systems come from? • Natural numbers • Whole numbers (Counting numbers) • Integers • Fractions (Rational numbers) • Roots and surds (Irrational numbers) • Properties of number systems (associative, commutative and distributive laws)
Exercise 1: Where do number systems come from? There is proof that the Ishango people of the Democratic Republic of the Congo (DRC) made marks on bones to count their cattle or family members. The oldest bone found to confirm these assumptions is approximately 20 000 years old. The number system that we use today is the Hindu-Arabic system and it was developed more than 1 000 years ago by Hindu-Arabic mathematicians. The Egyptian number system consists of symbols. Examples of the two number systems are: Egyptians
Hindu-Arabic
staff
1
heel
10
rope
100
lotus
1000
pointing finger
10 000
burbot
100 000
astonished man
1000 000
1.1*
Write down the Egyptian equivalent for the number 3 516.
1.2*
Write down the Egyptian equivalent for the number 2 182.
1.3***
Write the Hindu-Arabic number for
1.4*
Study the number 234 654 365 123 987 341 236 687. Write down the number that is 10 000 more than the given number.
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Study Guide G07 ~ Mathematics
1.5***
Unit
Investigate: For fun Draw the following table in your answer book and translate the terms addition, subtraction, multiplication and division in any other two languages. + English Afrikaans Sepedi
–
Addition Optelling Go hlakantšha
÷
x
Subtraction Aftrekking Go tloṧa
Multiplication Vermenigvuldiging Go atiša
Division Deling Go arola
Exercise 2: Natural numbers Natural numbers are the numbers from 1 to infinity. The symbol used is N. If tabulated: N = {1; 2; 3; 4; 5 …} If we look at all the number systems together, then A in this diagram will represent the natural numbers: The symbol for natural numbers is E N A represents natural numbers. As the unit progresses, the other number systems will be explained.
Units
7
Tens
1
Hundreds
Millions
3
Thousands
Ten millions
4
Ten thousands
Hundred millions
2
Hundred thousands
Milliards
Each digit in natural numbers has its own meaning. Example:
5
9
0
6
8
Milliard and billion denote the same number. Milliard is almost never used in America and Britain, but is often used in Continental Europe.
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Study Guide G07 ~ Mathematics
Unit
Decimal system
2.1**
1 million= 1000 000 (1000)2 = 106 Six zeros 1 billion A thousand = 1000 000 000 million = 109 Nine zeros 1 trillion (100 0000)2 =1000 000 000 000 =1012 Twelve zeros 1 quintillion (100 0000)3 = 1018 Eighteen zeros Complete the table by identifying the place value of the 7 .
Example
2.2**
Description Tens
2.1.1*
34 52 7 823
2.1.2*
2 7 82
2.1.3**
5 2 7 8 254
2.1.4**
7 254 164
2.1.5***
952 7 54 123
2.1.6*
1233452 7
2.1.7**
34 52 7 825 455
2.1.8*
7 82
2.1.9****
34 7 84 556 723
Write the number from the description given.
Example 2.2.1* 2.2.2* 2.2.3** 2.2.4* 2.2.5** 2.2.6*** 2.2.7****
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Number 345 2 7 8
Description 4 Hundred Thousands 6 Tens 12 Thousands 876 Millions 9 Units 47 Hundreds 639 Ten Thousands 32 456 Hundreds
5
Number 400 000
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Study Guide G07 ~ Mathematics
2.3***
Unit
Add 10 to all the following values. Write your answers in the table. Number 2.3.1* 65 382 2.3.2* 1 234 2.3.3** 87 592 2.3.4*** 96 795 2.3.5**** 9 328 999
2.4**
1
Answer
Complete the table by calculating what is asked.
Example 2.4.1** 2.4.2* 2.4.3** 2.4.4** 2.4.5**** 2.4.6**
Number 3 761 676 767 78 493 78 493 12 121 212 875 462 867 444 444
Calculation Add 400. Add 1 million. Subtract 50. Add 9 000. Subtract 1 million. Add four hundred thousand. Add 5 000.
Answer 4 161
2.5**
Break down the natural numbers as shown in the example. Number Answer Example 1 234 567 = 1000000 + 200000 + 30000 + 4000 + 500 + 60 + 7 2.5.1* 648 2.5.2* 33 333 2.5.3**** 54 545 454 2.5.4** 5 678 2.5.5* 6 2.5.6** 765 432
2.6**
Arrange the following natural numbers from small to big: {2542; 154; 2441; 2523; 2509}.
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Study Guide G07 ~ Mathematics
2.7**
Unit
1
Arrange the following natural numbers from big to small: {592; 523; 2600; 2699}. In Mathematics we use symbols to show greater than or less than. If you look at the following symbols from left to right, you can say:
>
greater than
<
Example: 234 > 233 345 < 346
2.8**
less than
Place a < or > symbol between the values in the table. Number 1 < or > Number 2 546 124 12 65 3 232 6 756 437 436 112 233 112 234 4 356 11 111
When doing addition and subtraction of natural numbers, your answer must have enough steps to show that you did not use a calculator. Example Make sure you know where the = 2 379 + 6 666 9 370 – 6 666 sign is placed. = 2 379 = 9 379 Know how and + 6 666 – 6 666 where this sign is 9 045 2 713 placed.
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Study Guide G07 ~ Mathematics
Unit
1
Simplify the following. Show your calculations. You are not allowed to use a calculator. 2.9*
17 + 25
2.10**
Use your answer and write down the correct answer for 17 hundred plus 25 hundred.
2.11**
Now use your answer and write down the correct answer for 17 million plus 25 million.
2.12****
Determine the sum of 17 Hundred Thousands and 25 Ten Thousands.
−
Calculations with natural numbers can only be one of the following:
+
Plus
×
Minus
Multiplication
÷ Division
Examples of calculations with natural numbers: Short division 25863 ÷ 8 =
3232 res 7 8 25863
Long division 25863 ÷ 8 3232 8 25863 = − 24 18 − 16 26 − 24 23 − 16 remainder 7
Adding 6765 + 25863 = 6765 + 25863 32628
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Multiplication 456 x 18 = 456 x 18 3648 + 4560 8208
Subtraction 25863 − 4444 = 25863 − 4444 21419
Study Guide G07 ~ Mathematics
Unit
1
Use the prescribed methods and determine the answers of the following. You are not allowed to use a calculator. No calculators. Write down all the actual calculations to show that you did not use a calculator.
2.13**
2 345 + 999
2.14**
2 345 – 999
2.15**
2 345 x 99
2.16***
2 345 ÷ 9 with the long division method.
2.17***
2 345 ÷ 99 with the long division method.
2.18****
2 345 ÷ 999 with the long division method.
Rewriting English sentences as mathematical sentences (number sentences) is essential to solving word problems. The following terms for calculations may be used. Study them and use them. Symbols Meaning Other English words Multiply Product of × Divide Quotient of ÷ – Subtract Difference between + Add Sum of Write the following English sentences in mathematical (number) sentences. Simplify them without using a calculator. The remainder is asked for question 2.21. That means that you have to do this question with long division. In future we will use decimals, but for now the remainder and quotient are both integers. 2.19**
Determine the sum of 12 and 56.
2.20**
Calculate the product of 12 and 56.
2.21**
What is the quotient if 56 is divided by 12? Give the remainder.
2.22***
A printing company prints 1 436 books per day. How many books did they print in July 2013 if they did not work on Saturdays and Sundays?
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Study Guide G07 ~ Mathematics
2.23****
Unit
1
A yacht sails in one direction. On the first day it sails 275 km and then 35 kilometres further each day. How far will the yacht be from the harbour at the end of the third day? Harbour
Day 1= 275 km Day 2 Day 3
2.24***
Juan drove 1 035 metres to a shop. John drove 2 285 metres to the same shop. How much further did John drive?
2.25
Sandra has to make 34 skirts for girls participating in a school play. She buys 2 rolls of material of 44 metres each. Each skirt requires 3 metres of material.
2.25.1***
Show all your calculations and determine how much material she still needs to buy. You are not allowed to use a calculator. 2.25.2*** To make the skirts, a single piece of 3 metres of material is required. The material can’t be joined. Explain how many rolls of material must be bought and how many metres will remain on each roll after 34 pieces of 3 metres have been cut. Rounding off natural numbers must be done In Tens, Hundreds and Thousands. Examples Question: Number 1 234 1 234 1 234 8 795 8 795 8 795
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Round off to the nearest … 10 100 1 000 10 100 1 000
Answer 1 230 1 200 1 000 8 800 8 800 9 000
10
Reason 4 is smaller than 5 34 is smaller than 50 234 is smaller than 500 5 is halfway to 10 95 is more than 50 795 is more than 500
Study Guide G07 ~ Mathematics
Unit
Rounding off to 10. (It means that the number must be divided by 10 without leaving a remainder.) Rounding off to 100. (It means that the number must be divided by 100 without leaving a remainder.)
Look at the Units. • When less than 5, round off downwards. • When 5 or more, round off upwards. Look at the Tens and Units (last two digits). • When less than 50, round off downwards. • When 50 or more, round off upwards Look at the last three digits (Hundreds, Tens and Units). • When less than 500, round off downwards. • When 500 or more, round off upwards.
Rounding off to 1 000. (It means that the number must be divided by 1 000 without leaving a remainder.)
2.26**
1
Complete the table by rounding off the numbers. Read the headings of the table.
Question: Number
To the nearest …
Answer
Reason
7 766 10 7 766 100 7 719 1 000 7 371 10 890 100 1 million 1 000 Round off the following numbers to the nearest 5. Example: (It means that 5 must be divided into the value without a remainder. Remember, the Units that are a 1 or a 2 must be rounded off downwards and Units that are 3 and 4, upwards.) • 62 rounded off to the nearest 5 is 60. • 63 rounded off to the nearest 5 is 65. • 84 rounded off to the nearest 5 is 85.
≈
When you round off in Mathematics, the following sign is used: For clarity, the number you round off to is written at the end of the answer. For examples: 26 ≈ 30 to the nearest 10 3 456 ≈ 3 500 to the nearest 100 2.27***
Round off 123 456 to the nearest 5.
2.28**** 2.29*** 2.29.1*
Round off 74 to the nearest 7. Shaun buys a pair of trousers for R243 and a jacket for R679. Round off each to the nearest R10 and give the total amount he paid for the trousers and the jacket. What will the amount be if Shaun first adds the amounts before rounding off? What is the difference between the answers in the previous two questions? Do you think one will always get to the same answers?
2.29.2** 2.29.3***
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Study Guide G07 ~ Mathematics
Unit
1
Order of operations When questions are asked that include a combination of adding, subtracting, multiplying or division, it is important to remember the rules regarding the order of operations.
Remember there are priorities for +, –, × and ÷
This sequence applies up to Grade 12. Study this now and you will never have to study it again.
The sequence of calculations is as follows: 1. Brackets (…) 2. Exponents. 3. Multiplication and division from left to the right. 4. Adding and subtraction from left to the right.
Simplify the following by doing only one calculation for each step: Example 1: 2+3x4 = 2 + 12 = 14
Example 2: 23 – (12 + 8) +2 = 23 – 20 + 2 =3+2 =5
Example 3: 30 ÷ 3 x 2 + 15 ÷ 5 = 10 x2 + 15 ÷ 5 = 20 + 15 ÷ 5 = 20 + 3 = 23
Example 4 2+3
10 – 6 + (3 – 2) +
=2+3
10 – 6 + 1 +
=2+3
10 – 6 + 1 + 6
of 12
Example: 5
of 12
2 + 23 = 2 + 23
= 2 + 30 – 6 + 1 + 6 = 32 – 6 + 1 + 6 = 26 + 1 + 6 = 27 + 6 = 33 Remember: Once the calculation in the bracket has been completed, the bracket disappears.
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2–
of (21 + 3)
2–
=2+8
2–
=2+8
2–8
= 2 + 16 – 8 = 18 – 8 = 10
of 24 of 24
Study Guide G07 ~ Mathematics
Unit
Simplify the following expression by doing one calculation per step only. 2.30* 4+2 3–1 Do one calculation for each step. 2.31* 1+3 2 6–2 It will teach you the preference and the use of the = sign. Remember 2.32** 9 – (12 – 8) that everything must be identical 2.33** 3+3 3–3 before and after the = sign. 2.34**
(3 + 2)
2.35**
2+2
2
2.36**
8–2
3 + 6 – (5 – 1)
2.37**
10 – 4 + (3 – 1)
2.38**
3×0 + 2×0 (6 – 2 + 7) + (3 – 2 + 12)
2.39** 2.40***
(6 – 2) 2 – 2 + (2 + 2)
Brackets first. A bracket disappears when the answer is written down.
3
Simplify by doing only one calculation for each step.
Treat the division line as two large brackets. • Simplify the numerator on its own and the denominator on its own. • Then divide the numerator by the denominator.
x and ÷ have the same priority. The first sign must be executed first.
Be aware
It is easy to multiply by 0. The answer stays zero.
All the number 2s can be confusing. Work systematically … one calculation for each step.
REMEMBER! (0)4 = 0 and not 4 6 x 0 = 0 and not 6
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Study Guide G07 ~ Mathematics
Unit
1
Factors This is a general term which means these are numbers that divide exactly into a bigger number, with an integer as an answer. Therefore 10 will be a factor of 20 because 10 can be divided into 20.
Prime factors A prime number is a number that has exactly two factors, namely the number 1 and the number itself. This means that only two numbers can be divided into a prime number without a remainder. There are an infinite number of prime numbers, but no formula exits to determine the number of prime numbers. The set of prime numbers is = {2; 3; 5; 7; 11; 13; 17; 19 â&#x20AC;Ś}
The number 1 1 is neither a prime number nor a composite number.
Composite factors Numbers that can be divided by more than 2 factors are called composite numbers. For example, 24 is composite because it can be divided by 1, 2, 3, 4, 6, 8, 12 and 24. Composite numbers have three or more factors. Example: Factors of 24 = {1; 2; 3; 4; 6; 8; 12; 24} = {the number 1} + {prime numbers} + {composite numbers} = {1} + {2; 3} + {4; 6; 8; 12; 24}
4 is written as 2 x 2 6 is written as 2 x 3 12 is written as 2 x 2 x 3 24 is written as 2 x 2 x 2 x 2 x 3
Prime numbers that are multiplied are called composite factors.
Write each number given as sets of 1, prime numbers and composite factors. Use the example given. 2.41*
12
2.42*
30
2.43**
36
2.44**
35
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First determine the prime factors.
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