Page 22 - 新思维数学学生用书7 样章

1.5 Tests for divisibility

• A number is divisible by 9 when the sum of the digits is divisible

by 9.
8 + 7 + 6 + 5 + 4 = 30 and 30 is not divisible by 9. So 87 654 is not
divisible by 9.
• A number is divisible by 10 when the last digit is 0.
The last digit of 87 654 is 4, so 87 654 is not divisible by 10.

• A number is divisible by 11 when the difference between the sum of
the odd digits and the sum of the even digits is 0 or a multiple of 11.
The sum of the odd digits of 87 654 is 4 + 6 + 8 = 18.

The sum of the even digits of 87 654 is 5 + 7 = 12.
18 − 12 = 6, so 87 654 is not a multiple of 11.

Worked example 1.9

The number *7 258 has one digit missing.
a Find the missing digit when:

i the number is divisible by 6 ii the number is divisible by 11
b A number is divisible by 66 when it is divisible by 6 and 11. Could *7 258 be divisible by 66?

Give a reason for your answer.

Answer
a i The number must be a multiple of 2 and 3.

The last digit is 8, so the number is divisible by 2.
The sum of the digits is * + 7 + 2 + 5 + 8 = * + 22.
If this is a multiple of 3, then * is 2 or 5 or 8.
There are three possible values for *.

ii The sum of the odd digits is 8 + 2 + * = 10 + *.
The sum of the even digits is 5 + 7 = 12.
When * = 2 the difference between these will be zero, so 27 258 is divisible by 11.
b The answer to part a shows that the number is divisible by 66 when * = 2. This is the only

possibility.

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