Manhattan Challenge Problem of the week ! - Jan 8

If x = 3 ^ 21 and y = 6 ^ 55, what is the remainder when xy is divided by 10?

(A) 2

(B) 3

(C) 4

(D) 6

(E) 8


Answer : The correct answer is E.

Since every multiple of 10 must end in zero, the remainder from dividing xy by 10 will be equal to the units’ digit of xy. In other words, the units’ digit will reflect by how much this number is greater than the nearest multiple of 10 and, thus, will be equal to the remainder from dividing by 10. Therefore, we can rephrase the question: “What is the units’ digit of xy?”

Next, let’s look for a pattern in the units’ digit of 3 ^ 21. Remember that the GMAT will not expect you to do sophisticated computations; therefore, if the exponent seems too large to compute, look for a shortcut by recognizing a pattern in the units' digits of the exponent:

3 ^ 1 = 3
3 ^ 2 = 9
3 ^ 3 = 27
3 ^ 4 = 81
3 ^ 5 = 243

As you can see, the pattern repeats every 4 terms, yielding the units digits of 3, 9, 7, and 1. Therefore, the exponents 3 ^ 1, 3 ^ 5, 3 ^ 9, 3 ^ 13, 3 ^ 17, and 3 ^ 21 will end in 3, and the units’ digit of 3 ^ 21 is 3.

Next, let’s determine the units’ digit of 6 ^ 55 by recognizing the pattern:

6 ^ 1 = 6
6 ^ 2 = 36
6 ^ 3 = 256
6 ^ 4 = 1,296

As shown above, all positive integer exponents of 6 have a units’ digit of 6. Therefore, the units' digit of 6 ^ 55 will also be 6.

Finally, since the units’ digit of 3 ^ 21 is 3 and the units’ digit of 6 ^ 55 is 6, the units' digit of 3 ^ 21 × 6 ^ 55 will be equal to 8, since 3 × 6 = 18.

Therefore, when this product is divided by 10, the remainder will be 8.