A number theory problem by Rmflute Shrivastav

Number Theory Level 2

( x + 1 ) 2 x + 23 \frac { {(x+1)} ^ 2 } { x + 23}

Find the largest integer x x for which this expression is also an integer.

461 463 462 450 460

Rmflute Shrivastav by Rmflute Shrivastav , 19, USA

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1 solution

Note first that ( x + 1 ) 2 x + 23 = x − 21 + 484 x + 23 . \dfrac{(x + 1)^{2}}{x + 23} = x - 21 + \dfrac{484}{x + 23}.

Now for the given expression to be an integer, we require that x + 23 x + 23 divide 484. 484. The largest such x x will be when x + 23 = 484 ⟹ x = 461 . x + 23 = 484 \Longrightarrow \boxed{x = 461}.

8 Helpful 0 Interesting 0 Brilliant 0 Confused

I did the same! A simple and efficient way to solve the problem.

tytan le nguyen - 5 years, 10 months ago

I just took the remainder and said

484 ≡ 0 ( m o d x + 23 ) ⇒ x + 23 = 484 , in k ( x + 23 ) = 484 , max ( x ) happens when k = 1 484 \equiv 0 (\mathrm{mod \; } x + 23) \\ \Rightarrow x + 23 = 484 \text{, in } k(x + 23) = 484\text{, max}(x)\text{ happens when k = 1}

Thus x = 461 \boxed{x = 461}

Kishore S. Shenoy - 5 years, 10 months ago

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