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1)If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n? a. 2 b. 3 c. 4 d. 6 e. 8
for the first one, suppose p = 4(any prime integer greater than 2, lets suppose 3)= 12
the dividers of 12 are: 1, 2, 3, 4, 6, and 12. altogather 6. so D is the answer.
2) If n is a positive integer and n^2 is divisible by 72, the largest +ve integer that must divide n is a. 6 b. 12 c. 24 d. 36 e. 48
for the second one, n^2 is a square of a positive integer and is also divisible by 72, whose factors are 2x2x2x3x3. so n could be any square number that is divisible by 72. for example 144, 144x4, 144x9, 144x16, 144x25, 144x36. remember n^2 cannot be 72 because n is a positive integer and sqrt(n^2) must be an integer.
lets see with some square values:
if n^2 = 144, n = 12 and n is divisible by 1,2,3,4,6, and 12. so 12 is the largest positive integer.
if n^2 = 144x4, n = 24 and n is divisible by 1,2,3,4,6,8, 12 and 24. so 24 is the largest positive integer. but n^2 could be 144 or 144x4. 24 works for the later one but doesnot work for the former where as 12 works for both so 12 or B is the correct answer.