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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 × 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 2b º b (mod 10) which implies that b = 0, which is clearly not a solution, as (1) would be false.

This implies c ³ 5. So we have 2b + 1 º b (mod 10) which implies b = 9.

As b ³ 5, we can also say 2a + 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