Problem Solving

Check this out when you have a minute and let me know if you know how to solve it properly...

If both 5^2 (25) and 3^3 (27) are factors of n x 25 x 62 x 73, what is the smallest possible positive value of n?

a) 25
b) 27
c) 45
d) 75
e) 125

Thanks!

IMO, the answer is (B). We're given that 27 is a factor of n x 25 x 62 x 73, which means that the factorization of n x 25 x 62 x 73 must include at least instances of 3 as a factor.

The number 62 is factored as 31x2, and can't go any further.
The number 73 is already prime, so no factorization possible.
The number 25, as we know, is 5x5.

We still need the three instances of 3 as a factor. The smallest number that provides these is 27.[/b]

ANSWER IS {D} - IF ANYONE CAN HELP WITH DETAILED METHODOLOGY AND WORK THAT WOULD BE GREAT!

CL3AV3R,

The answer D is incorrect. What's your source for this?

If we're looking for the smallest possible value of n, the only primes in n x 25 x 62 x 73 that we could overlook are the two fives in 25, since we know they're accounted for with the 5^2, which we know is a factor already.

The only factors left primes left are 2, 3, 3, 3, 31, and 73. And since we know 3^3 is a factor, 27 is smallest factor we can possibly create.

Answer: B

Hi,

Could be an error from the publisher

Answer in the back of the book is described as:

(d) The approach here is not to multiply out the numbers, but rather to completely factor the large number, then compare its factors to 52 and 33. Any 5s or 3s that can't be factored out of 25 x 62 x 73 (or 25 x 22 x 32 x 73) will have to be factors of n. Since we can account for two 3s, but no 5s in the large number, 52 and the remaining 3 must be factors of n.
Therefore n is equal to 5 x 5 x 3.[/spoiler]

*** Should read: then compare its factors to 5^2 and 3^3***[/b]

Hi, Anurag! Thanks for your time. Can you give me more detail about this question and take me through the step by step process broken down?? I am not sure if the incorrect answer is throwing me off or I can not remember how to solve this type of question...it has been a few years...

Thanks again![/b]

CL3AV3R wrote:Hi,

Could be an error from the publisher

Answer in the back of the book is described as:

(d) The approach here is not to multiply out the numbers, but rather to completely factor the large number, then compare its factors to 52 and 33. Any 5s or 3s that can't be factored out of 25 x 62 x 73 (or 25 x 22 x 32 x 73) will have to be factors of n. Since we can account for two 3s, but no 5s in the large number, 52 and the remaining 3 must be factors of n.
Therefore n is equal to 5 x 5 x 3.[/spoiler]

Looks as if there might have been a typo in your question up top, then. 52 and 33 don't appear in your question.

*** Should read: then compare its factors to 5^2 and 3^3***

wasn't showing exponents properly in that response...

I would go with B.

the multiple n x 25 x 62 x 73 contains no 3 - since 3^3 is a factor of the multiple, then n must contain at least 3^3

I go with B

Since none of the numbers are multiples of 27, n should be 27

(B) QED

25 & 33 are factors of "nx25x62x73" is given.

And from the question you can find out that 25 is already allocated in that number. Now look out for the 27 which is three 3's. There is no 3's in the "nx25x62x73", therefore the n"n" should be 27 in order to satisfy the factor

Answer is B

JohnQ2011 wrote:Check this out when you have a minute and let me know if you know how to solve it properly...

If both 5^2 (25) and 3^3 (27) are factors of n x 25 x 62 x 73, what is the smallest possible positive value of n?

a) 25
b) 27
c) 45
d) 75
e) 125

Thanks!

Misread it to say 5^2 * 25 and 3^3 * 27