Bunuel wrote:
Positive integer m is defined such that \(m=10^n−36\). Positive integer k is defined such that k equals the sum of the digits of integer m. For which of the following values of n is k a multiple of 5?
A. 107
B. 108
C. 109
D. 110
E. 111
Since m is a positive integer, we see that the least integer value of n is 2.
If n = 2, we see that m = 10^2 - 36 = 100 - 36 = 64.
If n = 3, we see that m = 10^3 - 36 = 1,000 - 36 = 964.
If n = 4, we see that m = 10^4 - 36 = 10,000 - 36 = 9,964.
If n = 5, we see that m = 10^5 - 36 = 100,000 - 36 = 99,964.
We see that if n > 2, m is the number consisting of a sequence of 9s followed by 64. In fact, the number of 9s is 2 less than n. For example, if n = 5, we have a sequence of three 9s followed by 64. Thus k, the sum of the digits of m, is the sum of these 9s and the digits 6 and 4, i.e., k = 9(n - 2) + 6 + 4 or k = 9(n - 2) + 10.
Since 10 is already a multiple of 5, in order for k to be a multiple of 5, 9(n - 2) also has to be a multiple of 5. Since 9 isn’t a multiple of 5, we know that (n - 2) must be a multiple of 5. This means that the units digit of n must be either 2 or 7. Of the answer choices given, only 107 has the desirable units digit, so choice A is the answer.
Answer: A
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