K = 10^26 + 2^26, and K is a multiple of 2^n… | GMAT Sprint

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PS · Problem Solving

705-805

Divisibility/Multiples/Factors

$K = 10^{26} + 2^{26}$ , and $K$ is a multiple of $2^n$ but NOT a multiple of $2^{n+1}$ . If n is a positive integer, what is the value of $n+1$ ?

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BrentGMATPrepNow

K = 10^{26} + 2^{26}

, and

K

is a multiple of

2^n

but NOT a multiple of

2^{n+1}

. If n is a positive integer, what is the value of

n+1

?

IMPORTANT: later in my solution, we need to recognize that

5^b

will end in 25 for all integer values of b greater than 0.

For example:

5^2 = 25

5^3 = 125

5^4 = 625

5^5 = 3125

5^6 = 15625

etc.

So:

K = 10^{26} + 2^{26} = 2^{26}(5^{26} + 1) = 2^{26}(XXXX25 + 1)

[aside: XXXX25 denotes some number ending in 25]…

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What this question tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Forgetting that 1 is not prime and that every integer divides 0.