Find a two digit number

Find number ab, where a is the tens' digit and b is the units’ digit, knowing that a⁴ + a² = 5 · b.

Solution:

The problem requires us to find a two digit number ab, where a is the tens digit and b is the units’ digit.

Figures may have values ​​from 1 to 9 (for a=0 result 0⁴ + 0² = 5 · b => b=0, and the number ab=0 is not a two-digit number).

From a⁴ + a² = 5 · b. We note that  5b is a multiple of 5.

a⁴ + a² = a² (a² +1) must be multiple of 5.

a²   can’t be 5 (a is a natural number)

a² +1 = 5; 

a² = 5-1 = 4; 

a = 2

4*5 = 5 *b

b = 4 

Other case, for the next multiple of 5

a² +1=10

a²=9

a=3

a²(a²+1) = 9*10 = 90 = 5*b =>b = 90/5 = 18 >9, impossible, b is a digit, b< 9

           

The only number ab that is solution to the problem is 24.