A 3-Digit Number

Number Theory Level 2

A 3-digt number (the first is not 0) a b c ‾ \overline { abc } have these properties:

a c b ‾ − a b c ‾ = 18 \overline { acb } - \overline { abc } =18

b a c ‾ − a b c ‾ = 180 \overline { bac } - \overline { abc } =180

What is the value of c b a ‾ − a b c ‾ \overline {cba } -\overline { abc } ?


The answer is 396.

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Tom Engelsman
Apr 4, 2021

We have the following expressions:

( 100 a + 10 c + b ) − ( 100 a + 10 b + c ) = 18 ⇒ 9 c − 9 b = 18 ⇒ c − b = 2 (100a+10c+b)-(100a+10b+c) = 18 \Rightarrow 9c-9b=18 \Rightarrow c-b=2 (i).

( 100 b + 10 a + c ) − ( 100 a + 10 b + c ) = 180 ⇒ 90 b − 90 a = 180 ⇒ b − a = 2 (100b+10a+c)-(100a+10b+c) = 180 \Rightarrow 90b-90a=180 \Rightarrow b-a=2 (ii).

The last computation gives us:

( 100 c + 10 b + a ) − ( 100 a + 10 b + c ) = 99 c − 99 a = 99 [ ( b + 2 ) − ( b − 2 ) ] = 99 ⋅ 4 = 396 . (100c+10b+a)-(100a+10b+c) = 99c - 99a = 99[(b+2) - (b-2)] = 99 \cdot 4 = \boxed{396}.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...