Divisibility tests

Does 8560 have factors from 2 to 10? We can tell without long division, just by looking at the end of the number.

• 10: the multiples of 10 (10, 20, 30, 40, ...) all end in 0. So a number is divisible by 10 if its last digit is 0. 8560 is; 125 is not.
• 5: the multiples of 5 end in 0 or 5, so that is the test for 5.
• 2: the multiples of 2 end in 0, 2, 4, 6 or 8, so that is the test for 2.

For 4, the last digit is not enough. 12 and 22 both end in 2, but 12 is a multiple of 4 and 22 is not. Instead, look at the number formed by the last two digits: if it is divisible by 4, the whole number is. For 8536, the last two digits make 36, and 36 = 4 × 9, so 8536 is divisible by 4.

For 8, look at the number formed by the last three digits: if it is divisible by 8, the whole number is. For 8560, the last three digits make 560 = 8 × 70, so 8560 is divisible by 8. (It is also divisible by 2, 4, 5 and 10.)

Two more tests you may meet later: a number is divisible by 3 if the sum of its digits is a multiple of 3, and by 9 if the sum of its digits is a multiple of 9. You can also break a divisor into co-prime parts: 15 = 3 × 5, so 1125 (it ends in 5, and its digits add up to 9) is divisible by 15. The parts must be co-prime: 12 is divisible by 2 and by 4, but not by 8.

These tests help with leap years. 2024 is a leap year. Leap years are the years that are multiples of 4, except the years divisible by 100 but not by 400. So 2100 will not be a leap year, but 2000 was one.