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Question

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Accepted Answer

Let x be the two-digit number, y be the three-digit number. Putting together the given, we have 1000x+y=9xy 9xy-1000x-y=0. Using SFFT, this factorizes to (9x-1)(y-10009)=10009, and (9x-1)(9y-1000)=1000. Since 89 < 9x-1 < 890, we can use trial and error on factors of 1000. If 9x - 1 = 100, we get a non-integer. If 9x - 1 = 125, we get x=14 and y=112, which satisifies the conditions. Hence the answer is 112 + 14 = 126.