Swapped Numbers

Number Theory Level 3

x y ‾ − y x ‾ = z \large \overline{xy}-\overline{yx}=z

For x > y x>y , find the smallest possible value of z z .

Notation: a b ‾ \overline{ab} denotes a two-digit number, where a a and b b are the ten digit and unit digit respectively.


The answer is 9.

Fidel Simanjuntak by Fidel Simanjuntak , 19, Indonesia

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2 solutions

x y ‾ − x y ‾ = 10 x + y − ( 10 y + x ) \overline{xy}-\overline{xy}=10x+y-(10y+x)

= 9 x − 9 y = z =9x-9y=z

Note that z z is a multiple of 9 9 . The smallest possible value of z z is 9 9

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Chew-Seong Cheong
Sep 18, 2016

z = x y ‾ − y x ‾ = 10 x + y − ( 10 y + x ) = 9 x − 9 y = 9 ( x − y ) Since x − y ≥ 1 ⟹ z ≥ 9 ( 1 ) = 9 \begin{aligned} z & = \overline{xy} - \overline{yx} \\ & = 10x+y - (10y+x) \\ & = 9x - 9y \\ & = 9(x-y) & \small \color{#3D99F6}{\text{Since }x-y \ge 1} \\ \implies z & \ge 9(1) = \boxed{9} \end{aligned}

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