A-two-digit-number-has-the-property-that-the-square-of-its-tens-digit-plus-ten-times-its-units-digit-is-equal-to-the-square-of-its-units-digit-plus-ten-times-its-tens-digit-Find-all-two-digit-numbers

Question Number 16935 by Tinkutara last updated on 28/Jun/17

A two - digit number has the property

that the square of its tens digit plus

ten times its units digit is equal to the

square of its units digit plus ten times

its tens digit . Find all two digit

numbers which have this property, and

are prime numbers .

Answered by RasheedSoomro last updated on 28/Jun/17

Let u is unit digit and t is tens digit .

t^{2} + 10 u = u^{2} + 10 t

t^{2} - u^{2} + 10 u - 10 t

(t - u) (t + u) - 10 (t - u) = 0

(t - u) (t + u - 10) = 0

t = u ∣ t = 10 - u

As the number is prime

u \neq 2, 4, 5, 6, 8

u may be 1, 3, 7 & 9

If t = u the numbers are 11, \overset{\times}{33}, \overset{\times}{77}, \overset{\times}{99}

If t = 10 - u the numbers may be

\overset{\times}{91}, 73, 37, 19

Total such numbers which have

mentioned property :

11, 33, 77, 99, 91, 73, 37, 19

Among which primes are :

11, 73, 37, 19 (Four)

Commented by Tinkutara last updated on 29/Jun/17

Thanks Sir!