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Answer to Question #539 in Combinatorics | Number Theory for teresa kellum

Question #539
Let 1210AB be a six-digit number with two digits (A and B) being unknown. Assume that the number shown is a multiple of both 9 and 11. Use the divisibility test for 9 to find the least positive residue of A + B (mod 9). Similarly, use the divisibility test for 11 to find the least positive residue of A – B (mod 11). Use these results to find the values of A and B that will make 1210AB a multiple of 99. Hint: A + B must be between 0 and 18, and A – B must be between –9 and 9.
Expert's answer
This number is a multiple of 11 => 1+1+A = (2+0+B) mod 11 => 1+1+A = 11k +2+0+B (k is any integer) => A-B = 11k. (A-B) must be between -9 and 9 => A-B = 0 => A = B. This number is a multiple of 9 => 1+2+1+0+A+B = 9k (k is any integer) => A+B = 9k - 4. A + B must be between 0 and 18 =>

1) A + B = 5 => 2A = 5 =>A=B = 2,5 (it's impossible, because A and B must be integer)
2) A + B = 14 => A = B = 7

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