OA = [spoiler][/spoiler]26[spoiler][/spoiler]

Let me put them as a,b,c for our notation. So a,b,c are different,non-zero single digit numbers now, ab * ba =cbca and a(a+c) = b a(a+c) = b we know that b is a single digit product of two single digits giving a single digit how many cases: 1* (1+{2,3,4,5,6,7,8,9} ) Hence a=1 - Let us keep this for some time.. another possibility.. 2*(2+1) because c cannot be 2 if a=2. [different digits] c cannot be 3 because b will become 10. so a=2, c=1 -> let us keep this also aside next: 3*(3+0) this is not possible because c cannot be 0. And if c is greater than 0 them b is more than 10. So for any value of "a" greater than 2, we will not have a feasible solution. So now a has to be either 1 or 2. when a=2, c=1, b= 6 coming to our other equation: ab * ba =cbca 26*62=1612 This is satisfied. Let us see what if a=1? Look at the last digit of the product -> b*a is yielding last digit as "a" if a=1, then last digit will be "b" and not "a" Hence a cannot equal 1. hence ab = 26

Here's how I did it. It's a bit unconventional, so looking fwd to better explanations! Rephrase problem as : =>xy * yx ____ zyzx We also have x(x+z) = y Start with this info to narrow down possibilities. 1. Since x,y and z are all different non-zero single digit integers, we can infer that y is a multiple of x, and since y cannot be equal to x, it must be 2x, 3x, etc. 2. Since minimum value of y is 2x and maximum value is 9, we can also infer that x < 5 since 2*5 = 10 which violates our single-digit constraint for y. So this leaves our possible values of x and y as : x = 1, y = 2,3,... x = 2, y = 4,6,8 x= 3, y = 6,9 x = 4, y = 8 Now we add our final constraint which is visible from the product. The units digit of the product is "x" so this means that the product of x*y must have a units digit of x. x= 1 will give units digit of 2,3,... but not 1. x = 2 gives us a units digit of 2 when y = 6 Test this to see if it satisfies our condition: 26*62 = 1612 => z = 1 which satisfies x(x+z) = y Pick 26.