Toy soldiers play the fool!

Algebra Level 2

I have a large number of toy soldiers, which I can arrange into a rectangle array consisting of rows and a number of columns. I notice that if I remove 100 toy soldiers, then I can arrange the remaining ones into a rectangle with 5 fewer rows and 5 more columns.

How many toy soldiers would I have to remove from the original configuration to be able to arrange the remaining ones into a rectangular array with 11 fewer rows and 11 more columns?

199 272 256 302 299 286

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

Let x x be the number of rows and y y be the number of columns. Then we are given that

( x − 5 ) ( y + 5 ) = x y − 100 ⟹ x y − 5 ( y − x ) − 25 = x y − 100 ⟹ 5 ( y − x ) = 75 ⟹ y − x = 15 (x - 5)(y + 5) = xy - 100 \Longrightarrow xy - 5(y - x) - 25 = xy - 100 \Longrightarrow 5(y - x) = 75 \Longrightarrow y - x = 15 .

Now let N N be the number of toy soldiers that must be removed to achieve the desired goal. This translates to satisfying the equation

( x − 11 ) ( y + 11 ) = x y − N ⟹ x y − 11 ( y − x ) − 121 = x y − N ⟹ 11 ( y − x ) = N − 121 ⟹ 11 × 15 = N − 121 ⟹ N = 286 (x - 11)(y + 11) = xy - N \Longrightarrow xy - 11(y - x) - 121 = xy - N \Longrightarrow 11(y - x) = N - 121 \Longrightarrow 11 \times 15 = N - 121 \Longrightarrow \boxed {N = 286} .

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