question 1 Find the smallest integer n such that n+1, n+2,and n+3 are all composites.

question 2 If n = 5! +1, show that n+1, n+2, and n+3 are all composite.

question 3 Find the sequence of 1000 consecutive composite numbers.

LEAH

Leah,

I did the first question by "guess and check".

2, 3, 4 - 2 is not composite
3, 4, 5 - 3 is not composite
etc

The first composite triple I found was 8, 9, 10. so n = 7.

When I saw the second question and n = 5! + 1 I went back to my answer to the first question and saw that n = 7 = 3! + 1. So, if n = 3! + 1 why are n + 1, n + 2 and n + 3 composites?

n + 1 = 3! + 2 = 3 2 1 + 2 = 2 (3 1 + 1) --- a composite
n + 2 = 3! + 3 = 3 2 1 + 3 = 3 (2 1 + 1) --- a composite
n + 3 = 3! + 4 = 3 2 1 + 4 = 2 (3 1 + 2) --- a composite

Now can you do questions 2 and 3?

Penny