Equation Solving

By Robert Cumming on Saturday, October 06, 2001 - 06:26 pm:

I am slightly stuck with this problem.

The letters a,b and c represent single digits.

Find all the possible sets of values of a,b and c given that:

abc + 2 =abb
abc x 2 =bba

(n.b. in this problem abc is the three digit number whose first digit is a, second b and third c. Similarily for abb and bba.)

By Maria Jose Leon-Sotelo Esteban on Sunday, October 07, 2001 - 10:36 am:

497+2=499
497+497=994
Maria Jose.

By Robert Cumming on Sunday, October 07, 2001 - 08:01 pm:

Thanks Maria

By James Myatt on Monday, October 08, 2001 - 05:51 pm:

Robert,

Look at the statements in detail.

(1) tells us that $c + 2 = b$ .

Consider the case with $c < 5$ . (2) implies $2 b \equiv b$ (mod 10) which implies that $b = 0$ , which is clearly not a solution, as (1) would be false.

This implies $c \geq 5$ . So we have $2 b + 1 \equiv b$ (mod 10) which implies $b = 9$ .

As $b \equiv 5$ , we can also say $2 a + 1 = b$ , so, using the statement above, $a = 4$ .

Similarly from (1), $c = 7$ , so the solution that Maria Jose has given you is the only solution to this problem.

Hope you understand,

James