Expert: Scott A Wilson - 4/8/2011

Question
the sum of the digits of a two digit number is 7 times the difference. the number itself exceeds 4 times the sum of the digits by 15. find the number?

Answer
The number is 10a + b, where a and b are the digits.

It is said that a+b = 7(a-b) or 7(b-a), depending on where a or b is larger.

The numer itself is 10a+b, and this is 4(a+b)+15.

Using 10a + b = 4a + 4b + 15, it can be seen that 6a = 3b + 15.
Dividing that by 3 gives [1] 2a = b + 5.  This say b must be odd,
for 5 is odd and b+5 must be even.

There are only 5 odd choices for b {1, 3, 5, 7, or 9}, so lets take the first as b = 1.  
This makes [1] into 2a = 1+5 = 6, as a = 3.  
The sum is 1+3=4, and that is not 7 times anything.

Let's try b = 3, so [1] says 2a = 3+5 = 8, so a = 4.
Note that 7(a-b) = 7, and a+b = 7, a+b = 7(a-b).
Also, 10a + b is 43, and 4(4+3)+15 = 4*7 + 15 = 28 + 15 = 43, so that equation is satisfied too.

The number given by a first and then b would be 43.