Number Properties

If the sum of n consecutive integers is 0, which of the following must be true? I. n is an even number II. n is an odd number III. The average (arithmetic mean) of the n integers is 0

If x and y are different prime numbers, each greater than 2, which of the following must be true? I. x + y ≠ 91 II. x - y is an even integer III. x/y is not an integer

If x*y = xy – 2(x + y) for all integers x and y, then 2* (–3) =

If x and y are prime numbers, which of the following CANNOT be the sum of x and y?

What is the least integer that is a sum of three different primes each greater than 20 ?

If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessary distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5. What is the greatest possible length of a positive integer less than 1000.

Which of the following must be equal to zero for all real numbers x ? I. − 1 x II. x + (-x) III. x^0

If r and s are integers and rs + r is odd, which of the following must be even ?

For how many integers n is 2^n = n^2 ?

If a = 1 and (a - b)/c = 1 which of the following is NOT a possible value of b?

Kim purchased n items from a catalog for $8each.Postageandhandlingchargesconsistedof$3 for the first item and $1 for each additional item. Which of the following gives the total dollar amount for Kim’s purchase, including postage and handling, in terms of n ?

For any integer n greater than 1, [n denotes the product of all the integers from 1 to n, inclusive. How many prime numbers are there between [6 + 2 and [6 + 6, inclusive?

For any positive integer n, n>1, the "length" of n is the number of positive primes (not necessarily distinct) whose product is n. For ex, the length of 50 is 3, since 50=2x5x5. Which of the following integers has length 3 ?

If a is a positive integer, and if the units digit of a^2 is 9 and the units digit of (a+1)^2 is 4, what is the units digit of (a+2)^2?

If positive integers x and y are not both odd, which of the following must be even?

How many three-digit numerals begin With a digit that represents a prime number and end with a digit that represents a prime number?

Which of the following integers does NOT have a divisor greater than 1 that is the square of an integer?

The average (arithmetic mean) of 3 different positive integers is 100 and the largest of these 3 integers is 120. What is the least possible value of the smallest of these 3 integers?

How many integers n greater than 10 and less than 100 are such that, if the digits of n are reversed, the resulting integer is n+9 ?