The product of the digits of a two digit positive number is 24. If 18 is added to the number then the digits of the number are interchanged. Find the number.
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let the units digit be x and tens digit be y
so the number is 10y + x
x * y = 24 ---> x = 24/y
10x + y - 18 = 10y + x
therefore
240/y + y - 18 = 10y + 24/y
240/y + y - 18 - 10y - 24/y = 0
216/y - 9y - 18 = 0
24/y - y - 2 = 0
y² + 2y - 24 = 0
y² + 6y - 4y - 24 = 0
y(y+6) - 4(y+6) = 0
(y-4)(y+6) = 0
therefore y = 4 or -6
but x and y are positive
therefore y = 4
x = 24/y = 6
therefore the number is 46
so the number is 10y + x
x * y = 24 ---> x = 24/y
10x + y - 18 = 10y + x
therefore
240/y + y - 18 = 10y + 24/y
240/y + y - 18 - 10y - 24/y = 0
216/y - 9y - 18 = 0
24/y - y - 2 = 0
y² + 2y - 24 = 0
y² + 6y - 4y - 24 = 0
y(y+6) - 4(y+6) = 0
(y-4)(y+6) = 0
therefore y = 4 or -6
but x and y are positive
therefore y = 4
x = 24/y = 6
therefore the number is 46
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