If p is a positive odd integer, what is the remainder when p is divided by 4 ?

(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

Of the 800 students at a certain college, 250 students live on campus and are more than 20 years old. How many of the 800 students live on campus and are 20 years old or less?

(1) 640 students at the college are more than 20 years old.
(2) 60 students at the college are 20 years old or less and live off campus.

If x is a positive integer, is x < 16 ?

(1) x is less than the average (arithmetic mean) of the first ten positive integers.
(2) x is the square of an integer.

If n and m are positive integers, what is the remainder when ~$3^{4n + 2 + m}$~ is divided by 10 ?

(1) n = 2
(2) m = 1

If y and z are integers, is y(z + 1) odd?

(1) y is odd
(2) z is even.

If k is an integer and 2 < k < 8, what is the value of k ?

(1) k is a factor of 30.
(2) k is a factor of 12.

Is the product of a certain pair of integers even?

(1) The sum of the integers is odd.
(2) One of the integers is even and the other is odd.

How many different prime numbers are factors of the positive integer n ?

(1) Four different prime numbers are factors of ~$2n$~.
(2) Four different prime numbers are factors of ~$n^2$~.

Of the students who eat in a certain cafeteria, each student either likes or dislikes lima beans and each student either likes or dislikes brussels sprouts. Of these students, ~$\frac{2}{3}$~dislike lima beans; and of those who dislike lima beans, ~$\frac{3}{5}$~also dislike brussels sprouts. How many of the students like brussels sprouts but dislike lima beans?

(1) 120 students eat in the cafeteria.
(2) 40 of the students like lima beans.

A garden store purchased a number of shovels and a number of rakes. If the cost of each shovel was $14 and the cost of each rake was $9, what was the total cost of the shovels and rakes purchased by the store?

(1) The ratio of the number of shovels to the number of rakes purchased by the store was 2 to 3.
(2) The total number of shovels and rakes purchased by the store was 50.