A two–digit number is such that the product of its digits is 35. If 18 is added to the number, the digit interchange their places. Find the number.
Let the two-digit number be xy(i.e. 10x + y).
After reversing the digits of the number xy, the new number becomes yx (i.e. 10y + x).
According to question –
xy = 35…..(1)
And,
(10x + y) + 18 = (10y + x)
⇒ 9x – 9y = – 18
⇒ x – y = – 2…..(2)
From equation(2), we get –
x = y – 2…..(3)
Substitute the value of x in equation(1), we get –
y(y – 2) = 35
⇒ y2 – 2y – 35 = 0
⇒ y2 – 7y + 5y – 35 = 0
⇒ y(y – 7) + 5(y – 7) = 0
⇒ (y – 7)(y + 5) = 0
∴ y = 7 [∵ y = – 5 is invalid because digit of a number can't be – ve.]
Substituting the value of y in equation (3), we get –
x = 5
Thus, the required number is 57.