Question: 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

A. Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are not sufficient.

Answer: C

Solution and Explanation:

Approach Solution 1:
Concentrate on the last digit of the number while calculating the remainder after dividing by 10. That will be all that is left. For instance, 85/10, remaining 5. 39 divided by 10, remainder 9.
Powers of three have a cyclicity of 4 in their final digit. Look at the illustration below:
3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
See the last digits: 3, 9, 7, 1, 3, 9, and so on. This is the pattern they follow. 36 = 729 and so forth.
Therefore, 3^{(4n + 2)} will result in 9. (If cyclicity makes you uncomfortable, look into its theory.
All that is left to know is what M is. 3^{(4n + 2)} + m will result in 0 if m = 1. Hence, the remainder will be 0.
C is the correct answer.
Correct Answer: C

Approach Solution 2:
Knowing n is pointless given that n and m are positive integers since the expression (4n+2) will always produce a result that is 4 units off. The value of m, which will determine the remainder of the statement when divided by 10, is what we need to know.
Keep in mind that the unit digit repeats itself in cycles of 4 for any exponent of 3:
3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
3⁶ = 729
3⁷ = 2187
3⁸ = 6561
Observe the pattern: 3-9-7-1
Hence, since m equals 1, we know that the exponent of the complete phrase (4n+2)+m will be 7, 11, 15, etc.
As a result, B is adequate, and the remaining will always be 7.
Correct Answer: C

Approach Solution 3:
Root Analysis:
3⁴ⁿ⁺² = (3⁴ⁿ)(3²)
Now that we have further broken this down, 9(34n)/10 + M/10. Now keep in mind that patterns are produced by bases to a power divided by the same denominator. Any number, PLUS a remainder of 1, will result from dividing any 34 power by 10 in this situation. Hence, 9(Q+1/10), where Q is an integer. So, we currently have 9Q + 9/10 + M/10. However, the value of M is yet unknown. We can answer the question if we know the value of M.
Proposition 1 According to our stem analysis, the value of N is useless; we require M. It's not enough.
Assertion 2 Because M is present and 9Q + 9/10 + 1/10 = 9Q + 1 suffices, the remaining value must be zero.
As a result,
Correct Answer: C

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