Prove that: (12^13–12^12+12^11)(11^9–11^8+11^7) is divisible by 3, 7, 19, and 37.
Wait so whats the answer
(12^13–12^12+12^11)(11^9–11^8+11^7)
= 12^11(12^2 - 12 + 1)(11^7)(11^2 - 11 + 1)
= 12^11(133)(11^7)(111)
= 12^11(7*19)(11^7)(3*37)
and we know that 12^11 divides by 3, since 12 is divisible by 3
all the other factors are plain to see
The answer is 12^11*11^7 I think
looking at numerator (by the way 12^ anything is divisible by 3 :)
one way
12^11 (144 -12 + 1) =12^11 (133)
hey look , 133 / 19 = 7 exactly so our mess is divisible by 19
12^10 ( 1728 - 144 + 12) = 12^10 (1596)
well, 1596 / 7 = 228 exactly
etc
Cool problem by the way :)
thanks:)
oh also try second term
(11^9–11^8+11^7) = 11^7 (11^2 - 11 + 1) = 11^7(111)
lol 111 = three times 37 Score !