Bunuel wrote:

If n is the remainder when 2^50 is divided by 3 and m is the remainder when 2^15 is divided by 5, what is m + n?

A. 6

B. 5

C. 4

D. 3

E. 2

Let’s determine the remainder pattern when 2 raised to an exponent is divided by 3.

(2^1)/3 = 0 remainder 2

(2^2)/3 = 4/3 = 1 remainder 1

(2^3)/3 = 8/3 = 2 remainder 2

(2^4)/3 = 16/3 = 5 remainder 1

When 2 is raised to an even exponent and is divided by 3, a remainder of 1 results. Thus, 2^50 divided by 3 has a remainder of 1.

Next we can determine the remainder when 2^15 is divided by 5. To do so, we recall that the units digit of any number divided by 5 will produce the same remainder as when the actual number is divided by 5. Thus, let’s determine the units digit of 2^15. The pattern of units digits of the base of 2 when raised to an exponent is:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 6

2^5 = 2

We see the pattern is 2-4-8-6, and furthermore that 2^4k, in which k is a positive integer, will always have a units digit of 6.

Thus, the units digit of 2^16 = 6 and so the units digit of 2^15 = 8. Dividing 8 by 5 yields a remainder of 3; thus, dividing 2^15 by 5 also yields a remainder of 3.

Therefore, m + n = 3 + 1 = 4.

Answer: C

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Founder and CEO

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