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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
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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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