Workload 3

Number Theory Level 3

A four-digit number of the form x x y y ‾ \overline{xxyy} , where x x and y y are single-digit positive integers , when divided by 11 gives a number which is again divisible by 11.

If x − y = 1 x - y = 1 , find the number.


The answer is 6655.

Achal Jain by Achal Jain , 19, India

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1 solution

Chew-Seong Cheong
Aug 17, 2016

x x y y ‾ ÷ 11 = x 0 y ‾ = 100 x + y Since x − y = 1 = 100 x + x − 1 = 101 x − 1 \begin{aligned} \overline{xxyy} \div 11 & = \overline{x0y} \\ & = 100x + \color{#3D99F6}{y} & \small \color{#3D99F6}{\text{Since }x-y=1} \\ & = 100x + \color{#3D99F6}{x-1} \\ & = 101x -1 \end{aligned}

We know that:

101 x − 1 ≡ 0 (mod 11) ⟹ 101 x ≡ 1 (mod 11) ( 99 + 2 ) x ≡ 1 (mod 11) 2 x ≡ 1 (mod 11) ⟹ x = 6 ⟹ y = 6 − 1 = 5 \begin{aligned} 101x -1 & \equiv 0 \text{ (mod 11)} \\ \implies 101x & \equiv 1 \text{ (mod 11)} \\ (99+2)x & \equiv 1 \text{ (mod 11)} \\ 2x & \equiv 1 \text{ (mod 11)} \\ \implies x & = 6 \\ \implies y & = 6-1=5 \end{aligned}

Therefore, x x y y ‾ = 6655 \overline{xxyy} = \boxed{6655}

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