for positive integer k, is the expression (k + 2)(k2 + 4k + 3) divisible by 4?
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(k+2)(k2+4k+3)=(k+1)(k+2)(k+3)(k+2)(k2+4k+3)=(k+1)(k+2)(k+3), so the expression is the product of three consecutive integers.
(1) k is divisible by 8 --> k=8n=evenk=8n=even --> (k+1)(k+2)(k+3)=odd∗even∗odd(k+1)(k+2)(k+3)=odd∗even∗odd. Now, k+2=8n+2k+2=8n+2, though even, is not a multiple of 4 (it's 2 greater than a multiple of 8), therefore the expression is not divisible by 4. Sufficient.
(2) (k + 1)/3 is an odd integer --> k+1=3∗odd=oddk+1=3∗odd=odd --> k=evenk=even --> (k+1)(k+2)(k+3)=odd∗even∗odd(k+1)(k+2)(k+3)=odd∗even∗odd. Now, k+2=evenk+2=even may or may not be divisible by 8, therefore the expression may or may not be divisible by 8. For example, consider k=2k=2and k=6k=6. Not sufficient.
Answer: A.
Hope it's clear
(1) k is divisible by 8 --> k=8n=evenk=8n=even --> (k+1)(k+2)(k+3)=odd∗even∗odd(k+1)(k+2)(k+3)=odd∗even∗odd. Now, k+2=8n+2k+2=8n+2, though even, is not a multiple of 4 (it's 2 greater than a multiple of 8), therefore the expression is not divisible by 4. Sufficient.
(2) (k + 1)/3 is an odd integer --> k+1=3∗odd=oddk+1=3∗odd=odd --> k=evenk=even --> (k+1)(k+2)(k+3)=odd∗even∗odd(k+1)(k+2)(k+3)=odd∗even∗odd. Now, k+2=evenk+2=even may or may not be divisible by 8, therefore the expression may or may not be divisible by 8. For example, consider k=2k=2and k=6k=6. Not sufficient.
Answer: A.
Hope it's clear
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