a) Show that the sum of 11 consecutive integers is always divisible by 11.
b) Show that the sum of 12 consecutive integers is never divisible by 12.
a): Label the integers: \(x-5, x-4, x-3, x-2, x-1, x, x+1, x+2, x+3, x+4, x+5=11x\) . And, this number is a multiple of \(11\) , so it is definitely divisible by \(11.\)
Try the second one on your own!
Thanks, tertre......here's another approach
Let the first integer = x
Let the 11th integer = x + 10
Sum = [ first integer + 11th integer] * number of terms / 2 =
[ x + x + 10 ] * 11 / 2 =
[2x + 10 ] * 11 / 2 =
2 [ x + 5 ] * 11 / 2 =
[ x + 5 ] * 11 which is divisible by 11