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Exercise 2C

1 Use the divisibility test for 3 to determine whether the following numbers are divisible by 3

87

2 Use the divisibility tests for 2 and 5 to determine whether the following calculations are possible without leaving a remainder.

3 Use the divisibility test for 9 to determine whether the following calculations are possible without leaving a remainder.

4 Use the divisibility test for 6 to determine whether the following numbers are divisible by 6 b Julie finds one more dollar on the floor and realises that she can now share the money equally among her three children. How much do they each receive?

5 Use the divisibility tests for 4 and 8 to determine whether the following calculations are possible without leaving a remainder.

6 Determine whether the following calculations are possible without leaving a remainder.

7 Carry out divisibility tests on the given numbers and fill in the table with ticks or crosses.

8 a Can Julie share $113 equally among her three children?

9 Write down five two-digit numbers that are divisible by:

10 The game of ‘clusters’ involves a group getting into smaller-sized groups as quickly as possible once a particular group size has been called out. If a year level consists of 88 students, which group sizes would ensure no students are left out of a group?

11 How many of the whole numbers between 1 and 250 inclusive are not divisible by 5?

12 How many two-digit numbers are divisible by both 2 and 3?

13 Blake is older than 20 but younger than 50. His age is divisible by 2, 3, 6 and 9. How old is he?

14 Find the largest three-digit number that is divisible by both 6 and 7.

REASONING

15 15, 16 16, 17

15 A number is divisible by 15 if it is divisible by 3 and by 5 a Determine if 3225 is divisible by 15, explaining why or why not. b Determine if 70 285 is divisible by 15, explaining why or why not. c The number 2 47 is divisible by 15. What could this number be? Try to list as many as possible. b Many of the digits in the number above can actually be ignored when calculating the digit sum. Which digits can be ignored and why? c To determine if the number above is divisible by 3, only five of the 21 digits actually need to be added together. Find this ‘reduced’ digit sum. a If the second-last digit is even, the last digit must be either a ____, ____ or ____. b If the second-last digit is odd, the last digit must be either a ____ or ____. b What is the difference between the two digits for each of these multiples? c Write down some three-digit multiples of 11. d What do you notice about the sum of the first digit and the last digit?

16 a Is the number 968 362 396 392 139 963 359 divisible by 3?

17 The divisibility test for the numeral 4 is to consider whether the number formed by the last two digits is a multiple of 4. Complete the following sentences to make a more detailed divisibility rule.

ENRICHMENT: Divisible by 11 ?

18 a Write down the first nine multiples of 11.

The following four-digit numbers are all divisible by 11: 1606, 2717, 6457, 9251, 9306 e Find the sum of the odd-placed digits and the sum of the even-placed digits. Then subtract the smaller sum from the larger. What do you notice? f Write down a divisibility rule for the number 11. g Which of the following numbers are divisible by 11?

An alternative method is to alternate adding and subtracting each of the digits. For example: 4 134 509 742 is divisible by 11

Alternately adding and subtracting the digits will give the following result: h Try this technique on some of your earlier numbers.