2.4: Divisibility Tests
An integer written in base 10:
- is divisible by 10 precisely when the units digit is 0.
Because 10 = 2 × 5, it follows that an integer (in base 10):
- is divisible by 5 precisely when the units digits is 0 or 5 (i.e. a multiple of 5); and
- is divisible by 2 precisely when the units digit is 0, 2, 4, 6, or 8 (i.e. a multiple of 2).
Because 100 = 4 × 25, it follows that an integer:
- is divisible by 4 precisely when the integer formed by its last two digits is a multiple of 4; and
- is divisible by 25 precisely when its last two digits are 00, 25, 50, or 75 (that is, a multiple of 25).
Because 1000 = 8 × 125, it follows that an integer:
- is divisible by 8 precisely when the integer formed by its last three digits is a multiple of 8.
Hence simple tests for divisibility by 2, by 4, by 5, by 8, and by 10 all follow easily from the way we write numbers in base 10.
- Prove that, when an integer is written in base 10, the remainder when it is divided by 9 is equal to the remainder when its “digit-sum” is divided by 9. Conclude that the remainder when an integer is divided by 3 is equal to the remainder when its “digit-sum” is divided by 3.
- Explain why an integer is divisible by 6 precisely when it is divisible both by 2 and by 3.
- What can you say about an integer N which is divisible by three times the sum of its base 10 digits?
- Find all integers which are equal to three times the sum of their base 10 digits.
- Find the smallest positive multiple of 9 with no odd digits.
Prove that an integer written in base 11 is divisible by ten precisely when its digit-sum is divisible by ten.