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What two-digit number is less than the sum of the square of

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What two-digit number is less than the sum of the square of [#permalink]  15 Nov 2012, 06:03
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What two-digit number is less than the sum of the square of its digits by 11 and exceeds their doubled product by 5?

(A) 95
(B) 99
(C) 26
(D) 73
(E) None of the Above
[Reveal] Spoiler: OA

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Re: What two-digit number [#permalink]  15 Nov 2012, 06:29
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vomhorizon wrote:
What two-digit number is less than the sum of the square of its digits by 11 and exceeds their doubled product by 5?

(A) 95
(B) 99
(C) 26
(D) 73
(E) None of the Above

We are told that the two-digit number exceeds doubled product of its digits by 5:

$$(10a+b)-2ab=5$$ --> $$2a(5-b)-(5-b)=0$$ --> $$(5-b)(2a-1)=0$$ --> $$b=5$$ ($$a$$ cannot equal to 1/2 since it must be an integer). The only answer choice with 5 as an units digit is A.

Check 95 for the first condition (to eliminate E), which says that the two-digit number is less than the sum of the square of its digits by 11: (9^2+5^2)-95=11. So, the answer is A.

There is another number satisfying both conditions:

Substitute $$b=5$$ in $$(a^2+b^2)-(10a+b)=11$$ --> $$a^2-10a+9=0$$ --> $$a=9$$ or $$a=1$$. Therefore both 15 and 95 satisfy both conditions.

Hope it's clear.
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Re: What two-digit number is less than the sum of the square of [#permalink]  12 Jul 2014, 05:29
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Re: What two-digit number is less than the sum of the square of [#permalink]  03 Sep 2014, 00:08
vomhorizon wrote:
What two-digit number is less than the sum of the square of its digits by 11 and exceeds their doubled product by 5?

(A) 95
(B) 99
(C) 26
(D) 73
(E) None of the Above

Let the digits be x and y. The number would be 10x + y.
We are given that 2xy + 5 = 10x +y = x^2 y^2 -11
Thus 2xy +5 = x^2 + y^2 - 11
x^2 + y^2 -2xy = 16
(x-y)^2 = 16
(x-y) = 4 or -4

Substituting the values of (x-y) in the equation 2xy +5 = 10x + y
x comes out to be 1 or 9... thus the two numbers can be 15 or 95
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Re: What two-digit number is less than the sum of the square of   [#permalink] 03 Sep 2014, 00:08
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