What is the remainder when the product of 100 5's are divided by 7?

Answer 1

The remainder is 2.

Knowing how to do modular arithmetic makes the problem much easier to solve. We say a number x has a value of k "mod" 7 if dividing x by 7 leaves a remainder of k. Multiples of 7 are equal to 0 mod 7 (they leave no remainder), and 32 is equal to 4 mod 7, because 32 / 7 = 4 remainder 4.

The product of 100 5s is equal to 5 to the 100th power, or 5^100 in shorthand. Let us look at what the first few powers of 5 are equal to mod 7:

5^1 = 5 = 5 mod 7

5^2 = 25 = 4 mod 7

5^3 = 125 = 6 mod 7

5^4 = 625 = 2 mod 7

5^5 = 3125 = 3 mod 7

5^6 = 15625 = 1 mod 7

5^7 = 78125 = 5 mod 7

5^8 = 390625 = 4 mod 7

It looks as if we might have a repeating pattern, and indeed the next few powers of 5 are equal to 6, 2, and 3, confirming that 5^k = 5^(k-6) mod 7. This means that 5^100 has the same remainder after dividing by 7 as 5^94, which has the same remainder as 5^88, etc. etc. until we reach 5^4, which leaves a remainder of 2. Therefore by induction 5^100 leaves a remainder of 2. Hope this all makes sense.