Problem Solving (PS) | Re: If m and n are positive integers and mn is a prime number, which of

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If\(m\) and\(n\) are positive integers and\(mn\) is a prime number, which of the following could be the units digit of\( 4^m +9^n\) ?

I. 3

II. 5

III.7


A. I only
B. II only
C. III only
D. I and II only
E. I and IIIonly


"\(mn\) is a prime number, implies that we can have the following fourcases:

Next, the units digit of 4 in positive integer power follows a repeating pattern of two digits: {4, 6} {4, 6} .... Hence, if\(m\) is odd, the units digit of\(4^m\) is 4; if\(m\) is even, the units digit of\(4^m\) is 6.

The units digit of 9 in positive integer power also follows a repeating pattern of two digits: {9, 1} {9, 1} .... Hence, if\(n\) is odd, the units digit of\(9^n\) is 9; if\(n\) is even, the units digit of\(9^n\) is1.

Thus:
[indent]
If\(m=1\) and\(n=2\) , then\( 4^m + 9^n = 4 + 81 =85\) , making the units digit equal to 5.

If\(m=1\) and\( n=odd \prime\) , then\( 4^m + 9^n = 4 + ...9 =...13\) , making the units digit equal to
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Statistics : Posted by Bunuel • on 25 Jun 2024, 01:30 • Replies 7 • Views 603

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