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Questions with division

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by TrueLie » Wed Mar 16, 2011 6:26 am
Dear all,

I do not understand the question, please explain it for me.

If n is a positive integer and n*n is divisible by 98, then the largest positive integer shown that must divide n is
(A) 2
(B) 7
(C) 14
(D) 28
(E) 56



Thank you very much.
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by force5 » Wed Mar 16, 2011 6:52 am
hi Again a classical question of prime factorization.
n^2 is divisible by 98 (7x7x2) ( given)
since n^2 is a perfect square (since n is given as an integer). it its prime box should have even powers of prime.

which means n2 cannot be 7x7x2 because 2 doesnt have even power. for it to be a perfect square we need to give it one more 2 which will make n^2 = 7x7x2x2 and hence n = 14....


hope this helped you.

thanks
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by 6983manish » Wed Mar 16, 2011 10:07 pm
force5 wrote:hi Again a classical question of prime factorization.
n^2 is divisible by 98 (7x7x2) ( given)
since n^2 is a perfect square (since n is given as an integer). it its prime box should have even powers of prime.

which means n2 cannot be 7x7x2 because 2 doesnt have even power. for it to be a perfect square we need to give it one more 2 which will make n^2 = 7x7x2x2 and hence n = 14....


hope this helped you.

thanks
What if we take n = 28 ,
n^2 = 784
n^2 / 98 = 784 / 98 = 8
( As question asks for the largest number and in the answer choices 28 seems largest fitting in the conditions )

We have 784 also divided by 98 and and its a square of 28.

and the largest number divides 28 is 28 itself.

Can we put answer as 28 ? Please comment.
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by TrueLie » Wed Mar 16, 2011 10:10 pm
Hi,

Thanks for your reply.
But the problem is that I did not understand the question: "then the largest positive integer shown that must divide n is "

Thank you.
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by Anurag@Gurome » Wed Mar 16, 2011 10:21 pm
TrueLie wrote:Hi,

Thanks for your reply.
But the problem is that I did not understand the question: "then the largest positive integer shown that must divide n is "

Thank you.
What they want to know is that for all possible values of 'n' from smallest to largest , such that n*n is divisible by 98, what is the largest integer that divides n.

This means that if the required integer is 'i', it has to divide the smallest value of 'n' also.

Since n^2 is divisible by 98, smallest possible value of n^2 is 196 and smallest value of n is 14.
So, i is always dividing 14 and its largest possible value is 14 itself.
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by TrueLie » Wed Mar 16, 2011 11:18 pm
Anurag@Gurome wrote:
TrueLie wrote:Hi,

Thanks for your reply.
But the problem is that I did not understand the question: "then the largest positive integer shown that must divide n is "

Thank you.
What they want to know is that for all possible values of 'n' from smallest to largest , such that n*n is divisible by 98, what is the largest integer that divides n.

This means that if the required integer is 'i', it has to divide the smallest value of 'n' also.

Since n^2 is divisible by 98, smallest possible value of n^2 is 196 and smallest value of n is 14.
So, i is always dividing 14 and its largest possible value is 14 itself.
Very clear answer, Anurag :)
Thank you very much.

However, n can be 28: 28*28 = 784 which is also divisible by 98.
Therefore, 'i' can be 28 or 56, right????
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by hoji » Thu Mar 17, 2011 2:33 pm
n^2 is divisible by 98=2*7*7
when solving this question, we must pay attention to the wordings

the largest positive integershown that must divide n,
in this case, we'll move from minimum to maximum:
1) assume that in order n^2 to be a perfect square the least integer we multiply to 98 is 2;
then n=14
2) next least integer is 8, then n=28,this can be the answer, not always: not qualifying "must"="always" part

so 3)....
answer can be 14, 28 or 56; but it must be 14
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