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in Olympiad by (65.8k points)

Consider all 6-digit numbers of the form abccba where b is odd. Determine the number of all such 6-digit numbers that are divisible by 7.

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Best answer

(abc cba) mod(7) = 0

=> (1 × a + 3 × b + 2 × c + 6 × c + 4 × b + 5 × a)

mod (7) = 0

(6a + 7b + 8c) mod (7) = 0

(7(a + b + c) + (c – a)) mod (7) = 0

(c – a) mod (7) = 0

=> c – a = 0 or c – a = 7 or c – a = –7

Numbers = 5 + 5 + 5 = 15

=> Required 6-digit numbers = 45 + 10 + 15 = 70

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