If k is a positive integer, what is the remainder when 62^k^3 is

Question

If k is a positive integer, what is the remainder when  62^{k^3}  is divided by 7?

(1) k is an even integer

(2) k^3 is divisible by 82

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zhanbo · Answer

Pre-thinking 
62 = 7*8+6
When 62 is divided by 7, the remainder is 6.
When 62^2 is divided by 7, the remainder is the same as 6^2 being divided by 7. The remainder is 1. 
Similarly, when 62^3 is divided by 7, the remainder is 6.
When 62^4 is divided by 7, the remainder is 1. 
While I could not prove this pattern, I have to believe that
The remainder is 6 when 62^n (n is odd positive integer) is divided by 7.
The remainder is 1 when 62^n (n is even positive integer) is divided by 7.

(1) Only
If k is even, k^3 is also even. The remainder is 6. 
Sufficient

(2) Only
If k^3 is divisible by 82, k^3 is even. 
Sufficient

Answer is  (D) .

rex199124 · Answer

This is how I did it,

62^(k^3)=(63-1)^(k^3)
All the terms of this expression contains 63 except for the last term, which contains some power of 1. So value of remainder depends on the value of (-1)^(k^3).
If K is even, (-1)^(k^3) = 1, reminder when divided by 7 is 1
If K is odd, (-1)^(k^3) = -1, reminder when divided by 7 is 6.

Statement 1: K is even, sufficient
Statement 2: k^3 is divisible by 82, which means k^3 is divisible by 2, so k^3 is even, which means K is even. Sufficient.

So the answer is D IMO

Amalgum23 · Answer

First, note:

62=56+6=7⋅8+6.

So when dividing by 7, 62 behaves the same as 6.

Also,

6^1 =6,
6^2=36=35+1.
Thus powers of 6 alternate:

odd exponent → remainder 6
even exponent → remainder 1

So all we need to know is whether k^3 is odd or even.

Statement (1): k is an even integer

If k is even, then k^3 is even.

Therefore 62^k^3 has an even exponent, so the remainder is 1.

Statement (1) is sufficient.

Statement (2): k^3 is divisible by 8

If k^3 is divisible by 8, then k must be even.

(An odd integer cubed remains odd and cannot be divisible by 8.)

Therefore k^3 is even, so the remainder is again 1.

Statement (2) is sufficient.

Since each statement alone is sufficient, hence D

If k is a positive integer, what is the remainder when 62^k^3 is

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