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.


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