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A pair of two-digit positive integers with reversed digits sum to a pe

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New post 12 Nov 2019, 23:18
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A pair of two-digit positive integers with reversed digits sum to a perfect square. How many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5
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Concentration: Finance, Operations
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A pair of two-digit positive integers with reversed digits sum to a pe  [#permalink]

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New post 12 Nov 2019, 23:49
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gracie wrote:
A pair of two-digit positive integers with reversed digits sum to a perfect square. How many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5


Ans. D

Explanation: Let the two digits be x and y.

N=10x+y;

Let M be the number obtained on reversing the digits.

M=10y+x;

N+M=10x+y+10y+x=11x+11y=11 (x+y);

N+M is a perfect square; i.e. 11 (x+y) is a perfect square; i.e. 11 divides the perfect square

The maximum value of N+M can be 99+99=198;

Consider the perfect squares series 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196. The next perfect square is 225. We need not go beyond 196 as the maximum value of N+M is 198.

In the above series of perfect squares, only 121 is divisible by 11.

Hence M+N=121 i.e. 11 (x+y)=121 i.e. x+y=11 and x, y are singe digit integers.

The integer solutions to the above equation are (2,9), (3,8), (4,7), (5,6).

Hence there are 4 such two digits numbers such that the sum of the number and its reverse is a perfect square. They are 29, 38, 47, 56.

Note: if repetition is included then there will be 8 pairs.

Please Give Kudos, If you find my answer good :thumbup: :thumbup:
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A pair of two-digit positive integers with reversed digits sum to a pe  [#permalink]

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New post 13 Nov 2019, 00:17
1
gracie wrote:
A pair of two-digit positive integers with reversed digits sum to a perfect square. How many such pairs are there?

A. 1
B. 2
C. 3
D. 4
E. 5

PQ= 10P +Q
QP=10Q+P
Sum=11(P+Q)
P+Q=11
(9,2),(8,3),(7,4),(6,5) Total 4 as (9,2) and(2,9) same pair only
D:)
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A pair of two-digit positive integers with reversed digits sum to a pe   [#permalink] 13 Nov 2019, 00:17
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