prove that if n is a positive integer then 24 dividesn+(n+1)(n+2).
Answers
Answer:n(n+1)(n+2) is the product of three consecutive integers because n is an integer.
0,1,2
-200,-199,-198
100,101,102
-1,0,1
In any set of three consecutive numbers, there must be at least one odd and one even.
odd,even,odd
OR
even,odd,even
A)even only when n is even
The product of three or more consecutive integers will always be EVEN. To make the product even, we just need one even. It really doesn't matter whether n is even or n+1.
If n is even, say 0
0,1,2. product=0=even
If n is odd, say -1
-1,0,1. product=0=even
Saying that n(n+1)(n+2) will be even ONLY if n=even is NOT correct.
B)even only when n is odd
We just saw that the product will always be even irrespective of whether n is even or odd.
C)odd whenever n is odd
Product will never be odd.
D)divisible by 3 only when n is odd
Rule: Product of n consecutive number will always be divisible be n!
{1,2}: Two numbers. n=2
1*2 will be divisible by 2!=2
{45,46,47,48,49,50}: Six numbers. n=6
45*46*47*48*49*50 will be divisible by 6!=720
Similarly,
3 consecutive numbers: {1,2,3}
1*2*3 will be divisible by 3!=6
If the product is divisible by 6, it must be divisible by its factor, which is 3.
Thus, "n" can be even/odd.
FALSE.
E)divisible by 4 whenever n is even
n=2
2,3,4. Product=24
TRUE.