Lesson: Introduction to Divisibility

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Whoa! Ok, I'm going to need a little more clarification here than most, if you don't mind.

N = 10^40 + 2^40. 2^k is a divisor of N, but 2^(k+1) is not a divisor of N. If k is a positive integer, what is the value of k−2?

gmat-admin's picture

Happy to help, bertyy!

You're referring to my question here: https://gmatclub.com/forum/n-10-40-2-40-2-k-is-a-divisor-of-n-237383.html

This is a VERY tricky question (750+)

I have a step-by-step solution here: https://gmatclub.com/forum/n-10-40-2-40-2-k-is-a-divisor-of-n-237383.htm...

Rather than repeat the entire solution, can you take a look and tell me which parts you'd like me to cllarify?

Hi Brent,

Factoring the first part is not the problem or seeing that "5" to the power of anything ends in five. The rest after that just throws me for a loop as to what to do or what is even going on. Usually I can force it after a while and come back to it, but this one has got me blocked. I can't figure out what i'm missing or that some concept that has escaped me.

Thanks for your help
gmat-admin's picture

Okay, I'll elaborate on my solution at https://gmatclub.com/forum/n-10-40-2-40-2-k-is-a-divisor-of-n-237383.htm...

WE have : N = 2^40(5^40 + 1)

We know that 5^40 must end in 25. So, we can say 5^40 = XXXX25 (the X's represent other digits on the number, but we don't really care about them.

So, we can say: N = = 2^40(XXXX25 + 1)

XXXX25 + 1 = XXXXX26 (if we add 1 to a number that ends in 25, the resulting value will end in 26

So, N = (2^40)(XXXX26)

At this part, we need to recognize that since XXX26 is EVEN. So, we can rewrite XXX26 as (2)(XXXX3).

So, N = (2^40)[(2)(XXXX3)]

We can now combine 2^40 and 2. We have: (2^40)(2) = (2^40)(2^1) = 2^41

So, N = (2^41)[XXXX3]

Since XXXX3 is an ODD number, we cannot factor any more 2's out of it. In other words, since XXXX3 is ODD, there are no 2's hiding in the prime factorization of XXXX3.

However, we DO know that there are 41 2's hiding in the prime factorization of 2^41. This means that 2^41 IS a factor of N, but 2^42 is NOT a factor of N.

The question tells us that 2^k is a factor of N, but 2^(k+1) is NOT a factor of N

In other words, k = 41

What is the value of k-2?

Since k = 41, we can conclude that k - 2 = 41 - 2 = 39

Does that help?


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