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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`
⇒ 10x2 + 16 - 160 - x2 = 54x
⇒ 9x2 - 54x - 144 = 0
⇒ x2 - 6x - 16 = 0
⇒ x2 - 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`
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