It can be shown (see exercises 1-4) that an integer is divisible by 3 if, and only if, the...

It can be shown (see exercises 1-4) that an integer is divisible by 3 if, and only if, the sum of its digits is divisible by 3. An integer is divisible by 9 if, and only if, the sum of its digits is divisible by 9. An integer is divisible by 5 if, and only if, its right-most digit is a 5 or a 0. And an integer is divisible by 4 if, and only if, the number formed by its right-most two digits is divisible by 4. Check the following integers for divisibility by 3, 4, 5 and 9.

a. 637,425,403,705,125

---

b. 12,858,306,120,312

---

c. 517,924,440,926,512

---

d. 14,328,083,360,232

Exercise 1

Prove that if n is any nonnegative integer whose decimal representation ends in 0, then 5 | n. (Hint: If the decimal representation of a nonnegative integer n ends in d0, then n = 10m + d0 for some integer m.)

Exercise 2

Prove that if n is any nonnegative integer whose decimal representation ends in 5, then 5 | n.

Exercise 3

Prove that if the decimal representation of a nonnegative integer n ends in d1d0 and if 4 | (10d1 + d0), then 4 | n. (Hint: If the decimal representation of a nonnegative integer n ends in d1 d0, then there is an integer s such that n = 100s + 10d1 + d0.)

Exercise 4

Observe that

Since the sum of the digits of 7524 is divisible by 9, 7524 can be written as a sum of two integers each of which is divisible by 9. It follows from exercise 15 that 7524 is divisible by 9.

Generalize the argument given in this example to any nonnegative integer n. In other words, prove that for any nonnegative integer n, if the sum of the digits of n is divisible by 9, then n is divisible by 9.

Exercise 5

Prove that for any nonnegative integer n, if the sum of the digits of n is divisible by 3, then n is divisible by 3.