Question

If from a two-digit number, we subtract the number formed by reversing its digits then the result so obtained is a perfect cube. How many such numbers are possible? Write all of them.

Answer 1

Let ab be any two-digit number.

Then, the digit formed by reversing it digits is ba.

Now, ab – ba = (10a + b) – (10b + a)

= (10a – a) + (b – 10b)

= 9a – 9b

= 9(a – b)

Further, since ab – ba is a perfect cube and is a multiple of 9.

∴ The possible value of a – b is 3.

i.e. a = b + 3

Here, b can take value from 0 to 6.

Hence, possible numbers are as follow.

For b = 0, a = 3, i.e. 30

For b = 1, a = 4, i.e. 41

For b = 2, a = 5, i.e. 52

For b = 3, a = 6, i.e. 63

For b = 4, a = 7, i.e. 74

For b = 5, a = 8, i.e. 85

For b = 6, a = 9, i.e. 96