#### Question

A two digit number is such that the product of the digits is 16. When 54 is subtracted from the number the digits are interchanged. Find the number

#### Solution

Let the two digits be:

Tens digits be x and units = 16/x

Number `=10x+16/x`

Number obtained by interchanging `=10xx16/x+x`

`rArr(10x+16/x)-(10xx16/x+x)=54`

`rArr10x+16/7-160/x+x=54`

⇒ 10x^{2} + 16 - 160 - x^{2} = 54x

⇒ 9x^{2} - 54x - 144 = 0

⇒ x^{2} - 6x - 16 = 0

⇒ x^{2} - 8x + 2x - 16 = 0

⇒ x(x - 8) + 2(x - 8) = 0

⇒ (x - 8)(x + 2) = 0

⇒ x - 8 = 0

⇒ x = 8

Or

⇒ x + 2 = 0

⇒ x = -2

But, a digit can never be negative, hence x = 8

Hence the required number `=10x+16/x=10(8)+16/8=80+2=82`

Is there an error in this question or solution?

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A Two Digit Number is Such that the Product of the Digits is 16. When 54 is Subtracted from the Number the Digits Are Interchanged. Find the Number Concept: Solutions of Quadratic Equations by Factorization.

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