Modular arithematic along with combi and permu!

Level 2

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

90 70 60 50

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1 solution

X X
Aug 29, 2018

a b c c b a \overline{abccba} is divisible by 7, so a b c c b a \overline{abc}-\overline{cba} is divisible by 7, and also a c a-c is divisible by 7.

b b has 5 5 possibilities, 1 , 3 , 5 , 7 , 9 1,3,5,7,9

( a , c ) (a,c) has 14 14 possibilities, a = c a=c has 9 9 possibilities, a c = 7 a-c=7 has 3 3 possibilities, c a = 7 c-a=7 has 2 2 possibities.

Hence 5 × 14 = 70 5\times14=70

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