Show that the integer n abcdef10 is divisible by 3 if and o

Show that the integer n = (abcdef)_10 is divisible by 3 if and only if (a + b + c + d + e + f) is divisible by 3.

Solution

writing the integer n as follows:

n = 10⁵a + 10⁴b + 10³c + 10²d + 10e + f

   = (99999+1)a + (9999+1)b + (999+1)c + (99+1)d + (9+1)e + f

   = (99999a + 9999b + 999c + 99d + 9e) + (a + b + c + d + e + f)

   = 3(33333a+3333b+333c + 33d + 3e) + (a + b + c + d + e + f)

clearly, the first term i.e. 3(33333a+3333b+333c + 33d + 3e) is divisible by 3.

So, for the integer n to be divisible by 3, the second term i.e. (a + b + c + d + e + f) must be divisible by 3.

Hence, the integer n is divisible by 3 if and only if (a + b + c + d + e + f) is divisible by 3.

(Proved)