A two digit number is obtained by multiplying the sum of the digits by 8. Also, it is obtained by multiplying the difference of the digits by 14 and adding 2. Find the number.

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Solution

Call the two-digit number xy, where x is the tens and y is the ones. Its value is 10x + y.
Then we can write
8(x + y) = 10x + y
and
14(x - y) + 2 = 10x + y
Simplifying these we get
8x + 8y = 10x + y

7y = 2x (or 14y = 4x)
and
14x - 14y + 2 = 10x + y
4x + 2 = 15y
Substitute from above and get
14y + 2 = 15y
y = 2
so then x = 7
Your number is 72.

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