Needgmat wrote:If n and m are positive integers, what is the remainder when 3^4n+2 + m is divided by 10?
1) n=2
2) m=1
If n and m are positive integers, what is the remainder when 3^(4n+2) + m is divided by 10?
We need to determine the remainder when 3^(4n+2) + m is divided by 10. Recall that if we know the last digit of a positive integer, then we know the reminder when that number is divided by 10 since the last digit of the number is the remainder. For example, when 2,016 is divided by 10, the remainder is 6 or when 27 is divided by 10, the remainder is 7.
Statement One Alone:
n = 2
Since n = 2, 4n + 2 = 10. So we need to determine the remainder when 3^10 + m is divided by 10. We can determine the remainder when 3^10 is divided by 10 (after all, 3^10 is a positive integer). However, since we don't know the value of m, we can't determine the remainder when 3^10 + m is divided by 10. Statement one is not sufficient to answer the question. Eliminate choices A and D.
Statement Two Alone:
m = 1
While the information in statement two seems insufficient to answer the question, it's actually sufficient. Recall that the remainder when a positive integer is divided by 10 is the same as the units digit. Let's review the units digits of powers of 3:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561
As we can see from the above, the units digits of powers of 3 have a pattern in a cycle of four: 3-9-7-1. Also, when this pattern of 4, we see that when 3 is raised to an exponent that is a multiple of 4, it will ALWAYS HAVE A UNITS DIGIT OF 1.
Thus, following our pattern, if n is a whole number, we know the following:
3^(4n) has a units digit of 1
3^(4n + 1) has a units digit of 3
3^(4n+2) has a units digit of 9
3^(4n + 3) has a units digit of 7
We see that 3^(4n+2) has a units digit of 9. If we add the value of m = 1 to that units digit, then the units digit of 3^(4n+2) + m will be 0. Statement two alone is sufficient to answer the question.
Answer:
B
