KA-0060Operation puzzles and cryptarithms
5 points, difficulty 3 of 3Level 5–6about 120sA and B are different digits. The two-digit number AB added to the two-digit number BA gives 132. What is A + B?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Write both numbers out in tens and ones.
2The strategy
AB means ten A plus B, and BA means ten B plus A. Add those two expressions and see what the total is a multiple of. The answer falls out without finding A and B separately.
3The full solution
(10A + B) + (10B + A) = 11A + 11B = 11(A + B). So 11(A + B) = 132, and A + B = 12.
Solution
The reliable way
Write each number in terms of its digits. AB is 10A + B and BA is 10B + A. Adding gives 11A + 11B, which is 11 times (A + B). So 11(A + B) = 132, and A + B = 132 divided by 11 = 12. You never need to know A and B separately — and in fact several pairs work, such as 48 and 84. The transferable idea: write the digits out algebraically and the structure appears.
The elegant way
Any two-digit number plus its reverse is 11 times the digit sum, so the answer is 132 divided by 11.
Why this is on the test: It rewards writing a digit puzzle out in tens and ones instead of guessing pairs, which is the technique every harder cryptarithm needs.