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VP  P
Joined: 07 Dec 2014
Posts: 1218
A pair of two-digit positive integers with reversed digits sum to a pe  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 42% (01:57) correct 58% (02:11) wrong based on 19 sessions

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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
Manager  S
Joined: 16 Feb 2015
Posts: 124
Location: United States
Concentration: Finance, Operations
A pair of two-digit positive integers with reversed digits sum to a pe  [#permalink]

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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  Manager  S
Joined: 10 Dec 2017
Posts: 150
Location: India
A pair of two-digit positive integers with reversed digits sum to a pe  [#permalink]

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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:) A pair of two-digit positive integers with reversed digits sum to a pe   [#permalink] 13 Nov 2019, 00:17
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