Question

A two digit number is such that the product of its digits is 16 when 72 is added to the number the digits interchange their places. Find the number

Answer 1

Let this two digit number be xy.

We can write equations from the information we have on this digit.

Given product of its digit is 16, then (xy) = 16 … (1)

∴ x = 16/y and y = 16/x

The number can also be written as (10x + y) 

Given that when 72 is added to the number the digits gets interchanged.

Hence 10x + y + 72 = 10y + x

9x – 9y + 72 = 0

x– y + 8  = 0 … (2)..............this is the second equation

∴ x= y - 8 

From (1) and (2), we get

We can substitute y =16/x in equation (2)

x – (16/x) + 8 = 0

x²- 16 + 8x = 0

x² + 8x - 16 = 0

The solution to this equation does not give a whole number, hence xy is an impossible two digit number. 

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Here is an example of a similar question that can help you get the concept:

Q. A two digit number is such that the products of its digits is 14, when 45 is added to the number digits interchange their place. find the number?

A.  xy = 14 … (1)
∴ y = 14/x

The number is (10x + y)

When 45 is added to the number the digits gets interchanged. Hence:

 10x + y + 45 = 10y + x

9x – 9y + 45 = 0

x– y + 5 = 0 … (2)

Substitute equation 1 in equation 2.

x – (14/x) + 5 = 0

x² + 5x – 14 = 0

x2 + 7x – 2x – 14 = 0

x(x + 7) – 2(x + 7) = 0

(x + 7)(x – 2) = 0

x = - 7 or x = 2

∴ x = 2

Find y; y = 14/x  y = 14/2  = 7

∴y = 7

The number is (10x + y) = 27

∴ The two digit number is 27