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n is an integer and n

I. n

II. n is divisible by 4.

III. n

^{2}is divisible by 28. Which of the following must be true?I. n

^{2}is divisible by 49.II. n is divisible by 4.

III. n

^{2}/28 is divisible by 7.Correct Answer:

**C**The equation for an arithmetic sequence is a

_{n}= a_{1}+ d(n-1)- It is important to realize that given the statement that n is an integer, it is impossible for n
^{2}to have only one of any individual factor. For example if n were 6 = 3*2, then n^{2}would be 36 = 2*2*3*3. In other words, the number of occurrences of each individual factor is doubled. - n
^{2}is a multiple of 28 so we know that n^{2}has at least one factor of 7 and two factors of 2 (28 = 7*2*2). We also know from the above paragraph that n^{2}cannot just have one factor of 7, therefore n^{2}has at least two sevens in its factor tree. - I. n
^{2}has at least two factors of 7, so n^{2}must have a factor of 49.

II. n^{2}could be 7*7*2*2 = 196 which would make n = 14. This is not divisible by 4.

III. n^{2}has at least two factors of 7, so when you divide by 28, a number with only one factor of 7, at least one factor of 7 remains. Therefore n^{2}/28 is divisible by 7. - I and III are correct so the answer is C.