Can you solve Sarah's number?

Algebra Level pending

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?

126 421 127 342 876

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Apr 30, 2017

Let x x be the two digit number and y y the three digit number. Putting together the given, we have 1000 x + y = 9 x y ⟹ 9 x y − 1000 x − y = 0 1000x+ y= 9xy \implies 9xy-1000x-y=0

This can be factored into ( 9 x − 1 ) ( y − 1000 9 ) = 1000 9 (9x-1)(y-\frac{1000}{9}) = \frac{1000}{9} , and ( 9 x − 1 ) ( 9 y − 1000 ) = 1000 (9x-1)(9y-1000) =1000 .

Since 89 < 9 x − 1 < 890 89<9x-1<890 .

If 9 x − 1 = 100 9x-1=100 , it doesn't work.

If 9 x − 1 = 125 9x-1= 125 , we get x = 14 x=14 and y = 112 y=112 which satisfies our conditions.

Thus, the answer is 112 + 14 = 126 112+14= \boxed{126} .

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