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Question

The sum of digits of a two digit number is 7. If the digits are reversed and the resulting number is decreased by 2, twice the original number is obtained. Find the original number

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Solution

Let the original number be ¯¯¯¯¯¯yx; i.e., 10y+x. We know x+y=7. The number obtained by reversing the digits in ¯¯¯¯¯¯xy, i.e., 10x+y. The second condition gives 10x+y2=2(10y+x). Thus we have two equations:
x+y=7....(1)
8x19y=2....(2)
Multiply the equation (1) by 19 and get
19x+19y=133.
Adding this to (2), we obtain 27x=135. This gives x=5. Hence y=7x=75=2.
The required number is 25.

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