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 .

Step-by-step solution

abccba ( b is odd) aligned & =a (10^5+1 )+b (10^4+10 )+c (10^3+10^2 ) & =a(1001-1) 100+a+10 b(1001)+(100)(11) c & =(7.11 .13 .100) a-99 a+10 b(7.11 .13)+(98+2)(11) c & =7 p+(c-a) where p is an integer aligned Now if c - a is a multiple of 7 c-a=7,0,-7 Hence number of ordered pairs of (a, c) is 14 since b is odd, Number of such number =14 5=70