Number system

If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

For positive integers x and y, which of the following can be written as y^2?

What are the last two digits of (301*402*503*604*646*547*448*349)^2 ?

A set of five positive integers has an arithmetic mean of 150. A particular number among the five exceeds another by 100. The rest of the three numbers lie between these two numbers and are equal. How many different values can the largest number among the five take?

Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x-1, then which of the following expressions must be true?

s(n) is a n-digit number formed by attaching the first n perfect squares, in order, into one integer. For example, s(1) = 1, s(2) = 14, s(3) = 149, s(4) = 14916, s(5) = 1491625, etc. How many digits are in s(99)?

If x is an odd negative integer and y is an even integer, which of the following statements must be true?
I. (3x - 2y) is odd
II. xy^2 is an even negative integer
III. (y^2 - x) is an odd negative integer

A “Sophie Germain” prime is any positive prime number p for which 2p + 1 is also prime. The product of all the possible units digits of Sophie Germain primes greater than 5 is

The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?

How many positive even integers less than 100 contain digits 4 or 7?

Two integers x and y are chosen without replacement out of the set {1, 2, 3,......, 10}. Then the probability that x^y is a single digit number.

How many positive integers less than 20 can be expressed as the sum of a positive multiple of 2 and a positive multiple of 3?

How many prime numbers exist between 200 and 220?

If 3 < x < 100, for how many values of x is x/3 the square of a prime number?

An integer between 1 and 300, inclusive, is chosen at random. What is the probability that the integer so chosen equals an integer raised to an exponent that is an integer greater than 1?

If two integers are chosen at random out of the set {2, 5, 7, 8}, what is the probability that their product will be of the form a^2 – b^2, where a and b are both positive integers?

Consider a sequence of numbers given by the expression 5 + (n - 1) * 3, where n runs from 1 to 85. How many of these numbers are divisible by 7?

If x/(11p) is an odd prime number, where x is a positive integer and p is a prime number, what is the least value of x?

When a certain perfect square is increased by 148, the result is another perfect square. What is the value of the original perfect square?

If, for all positive integer values of n, P(n) is defined as the sum of the smallest n prime numbers, then which of the following quantities are odd integers?

I. P(10)
II. P(P(10))
III. P(P(P(10)))