Both

**a** and

**b** are four-digit numbers and one is obtained from
the other by reversing the digits.

Determine them, knowing that

**2 * (a + b) = 5 * (b - 1)**.

Show your reasoning.

The problem asks for a and b such that 2a=3b-5. Obviously, b must be odd so 3b-5 is even. Writing a=1000w+100x+10y+z we get 2000w+200x+20y+2z= 3000z+300y+30x+3w-5. Equating the larger terms, 2w=3z, which allows for w=3 or w=9 since w must be odd so b is odd.

If w=3, z=2, and the equality implies 170x=280y, so x=y=0, producing a=3002 and b=2003.

If w=9, z=6, and we et 170x=280y+10, so x=5 and y=3, giving a=9536 and b=6359.