Divisibility Tests 5.2

Question 1

When is a number divisible by 10?

Answer 1

If and only if its last digit is 0

Question 2

When is a number divisible by 5?

Answer 2

If and only if its last digit is 0 or 5

Question 3

When is a number divisible by 2?

Answer 3

If and only if its last digit is 0, 2, 4, 6, or 8

Question 4

When is a number divisible by 4?

Answer 4

If and only if its last two digits are a number divisible by 4

Question 5

When is a number divisible by 8?

Answer 5

If and only if its last three digits are a number divisible by 8

Question 6

When is a number divisible by 3?

Answer 6

If and only if the sum of the digits is divisible by 3

Question 7

When is a number divisible by 9?

Answer 7

If and only if the sum of the digits is divisible by 9

Question 8

When is a number divisible by 11?

Answer 8

If and only if the number formed by the (sum of odd-position digits) - (sum of even-position digits) is a multiple (positive, negative or zero) of 11

Question 9

What is the proof of the divisibility test for 9?

Answer 9

Consider a three-digit number N with digits abc (the proof with more digits is similar). In expanded form, N = 100a + 10b + c. Note that 100a = 99a + a and 10b = 9b + b. So we can rewrite this as N = (99a + 9b) + (a + b + c). Now (99a + 9b) = 9(11a + b) is divisible by 9 . By the Divisibility Lemma, N is divisible by 9 if and only if a + b + c is divisible by 0. But a + b + c is the sum of the digits.

Question 10

What is the Divisibility Lemma (Lemma 2.5)?

Answer 10

Suppose A is a number divisible by k. Then B is divisible by k A = B is divisible by k.