Question

The sum of both digits, of either of two two-digit numbers , in whatever order the digits are written, is 9. The square of either of the digits of either number, minus the product of both digits, plus the square of the other digit is the number 21. The numbers are?

a. 36, 63
b. 81, 18
c. 27, 72
d. 45, 54
e. none

Answer 1

(d.) 45, 54

Explanation:

Let the first digit = y

Let the second digit = z

y +z = 9 ------------------------------------------ (1)

y²- yz +z² = 21------------------------------------(2)

From equation (2),

z = 9-y-------------------------------------------------(3)

Substitute equation (3) into (2):

y²- y(9-y) +(9-y)² = 21

y²-9y+y²+y²-18y+81 = 21

3y²-27y+ 81 = 21

3y²-27y+ 81-21= 0

3y²-27y+ 60= 0

y²- 9y +20= 0

(y -5) (y-4) =0

y= 5 or y =4

z = 4 or 5 (substituting into (3))

So the numbers are 54 or 45.