Number Theory problem

For how many positive integral values of N is the expression N(N-101) the perfect square of a positive integer ?

I got 2601 as the only value. I want to know if there are other such values ? I know it's a silly question for Brilliant problem solvers. Please help.........

Note by Nishant Sharma
6 years, 4 months ago

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4 votes

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Let x2=Nx^2=N and

y2=N101y^2=N-101 where x>0,y>0x>0,y>0

Now we can write from our first statement that :

y2=x2101y^2=x^2-101

x2y2=101\Rightarrow x^2-y^2=101

(x+y)(xy)=101(x+y)(x-y)=101

Therefore it follows that (x+y)(x+y) and (xy)(x-y) are factors of 101101.

Now since 101101 is a prime and x,yx,y are positive integers then the only possible values that (x+y)(x+y) can take is 101101.

Note that since x>yx>y ,, (xy)101(x-y) \neq 101.

So the only solution possible is when x=51x=51 which gives y=50y=50.

Therefore N=x2=2601N=x^2=2601 is the only solution.

Aditya Parson - 6 years, 4 months ago

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You need to substantiate your first line. Why must we have N=x2N=x^2? You are merely given that NN is an integer, and not told that it is a square. For example, if we ignore the condition that N(N101) N(N-101) must be positive, then N=101N=101 will be a solution, but that disagrees with your claim that NN must be a perfect square.

Calvin Lin Staff - 6 years, 4 months ago

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If N=101N=101 is a solution then y=0y=0 but I already stated that y>0y>0. So N=101 cannot be a solution.

Aditya Parson - 6 years, 4 months ago

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it came in isi right

superman son - 6 years, 4 months ago

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yea,you too gave isi,how many could you answer in part i & II...?

Sayan Chaudhuri - 6 years, 4 months ago

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Yeah it was an ISI problem. Well I attempted 29/30 in part I and solved 4 in part II. How about you both ?

Nishant Sharma - 6 years, 4 months ago

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@Nishant Sharma i am 13 so i cannot participate but i saw the question and tried to solve it . i got about 28 in part 1 and 7 in part 2

superman son - 6 years, 4 months ago

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@Superman Son Are you sure or just kidding(your age) ? Anyway, where from did you get the questions ?

Nishant Sharma - 6 years, 4 months ago

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@Nishant Sharma y kiddin and q from http://cheentaganitkendra.blogspot.in/2013/05/isi-2013-bmath-bstat-subjective-paper.html

superman son - 6 years, 4 months ago

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You are missing another obvious (positive integral) solution. (This statement is false. See edit below.)

Hint: Greatest Common Divisor.

Edit: I missed out that N(N101)N (N-101) has to be the square of a positive integer. I was thinking that N=101N=101 works.

Calvin Lin Staff - 6 years, 4 months ago

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I don't think that any other solution exists, Calvin Sir. Both NN and N(N101)N(N-101) are to be positive.

Shourya Pandey - 6 years, 4 months ago

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I misread, didn't see "positive" in the latter instance. Thanks for correcting.

Calvin Lin Staff - 6 years, 4 months ago

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Me too. That was why I posted this since I had done few problems only. BTW did you also write ISI 2013 ?

Nishant Sharma - 6 years, 4 months ago

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yes

superman son - 6 years, 4 months ago

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Seriously could not follow you. Please extrapolate a bit.

Nishant Sharma - 6 years, 4 months ago

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