Question

A two-digit number contains the smaller of the two digits in the unit place. The product of the digits is 40 and the difference between the digits is 3. Find the number.

Answer 1

Let the digit in the unit place be $$x$$. 

Since the unit place has the smaller digit and the difference between digits is 3, the ten’s digit is $$(x + 3)$$.

According to the condition: $$x(x + 3) = 40$$

$$x^2 + 3x - 40 = 0$$

Factoring: $$x^2 + 8x - 5x - 40 = 0$$

$$x(x + 8) - 5(x + 8) = 0$$

$$(x + 8)(x - 5) = 0$$

$$x = -8 \quad \text{or} \quad x = 5$$

Since a digit cannot be negative, $$x = 5$$.

Therefore, unit’s digit $$= 5$$ and ten’s digit $$= 5 + 3 = 8$$.

Hence, the number is $$10(8) + 5 = 85$$.