GMAT Math : Properties of Integers
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Example Questions
Example Question #21 : Understanding The Properties Of Integers
What is the product of all of the prime numbers between 20 and 30?
There are two prime numbers between 20 and 30; they are 23 and 29. Their product:
Example Question #181 : Arithmetic
What is the sum of all of the composite numbers from 31 to 40 inclusive?
The composite numbers that fall in the range 31-40 - those with more than two factors - are 32, 33, 34, 35, 36, 38, 39, and 40. Add them:
Example Question #23 : Understanding The Properties Of Integers
Which of the following integers cannot be the product of two distinct prime numbers?
We look at the prime factorizations of each of the five choices, and see which one breaks down as the product of more than two primes - this is the correct choice.
None of the above choices are correct, as each can be written as the product of two primes. As for the other choice:
Therefore, it cannot be expressed as the product of two primes. This is the correct choice.
Example Question #24 : Understanding The Properties Of Integers
A one-hundred digit integer that is divisible by 12 has 1's as its first 98 digits. Which of the following could be its last two digits, in order?
For a number to be divisible by 12, it must be divisible by 3 and 4.
For a number to be divisible by 4, the last two digits must form a number divisible by 4. This allows us to eliminate 34 and 54.
For a number to be divisible by 3, the digit sum must be divisible by 3. The sum of the first 98 digits, all of which are 1's, is 98; we add this 98 to the sum of each remaining pair of digits to determine which passes this test:
64 is the correct choice.
Example Question #21 : Properties Of Integers
Add the squares of the prime numbers between 50 and 60.
There are two primes between 50 and 60 - 53 and 59. The sum of their squares is:
Example Question #26 : Understanding The Properties Of Integers
What are the last two digits, in order, of 
Inspect the first few powers of 15; a pattern emerges.
After the first power, any odd power of 15 ends in the digits 75. Since 1,515 is odd, this is the correct choice.
Example Question #27 : Understanding The Properties Of Integers
Each box in this five-digit number is to be replaced by a digit to form a number divisible by 3:
How many ways can this be done using the same digit in each box?
A number that is divisible by 3 has digit sum that is divisible by 3. Let 
Substitute each of the integers from 0 to 9 for 
Therefore, there are three ways to fill the box in with the same digit to make the number a divisible by 3.
Example Question #22 : Properties Of Integers
What is the sum of the composite numbers between 1 and 100 inclusive that have 7 as their last digit?
The composite numbers in the 1-100 range that end in 7 are 27, 57, 77, and 87.
Add:
Example Question #23 : Properties Of Integers
Each box in this five-digit number is to be replaced by a digit to form a number divisible by 4:
How many ways can this be done using the same digit in each box?
Whether the integer is divisible by 4 is determined by the last two digits - the number is divisible by 4 if and only if the last two digits form a number itself divisible by 4. There are five digts that you can put in the box to achieve this - 1, 3, 5, 7, 9 (16, 36, 56, 76, and 96 all are evenly divisible by 4).
Example Question #26 : Understanding The Properties Of Integers
Multiply all of the composite integers between 1 and 100 that have 3 as their last digit. What is the product?
The composite numbers in the 1-100 range that end in 3 are 33, 63, and 93. Multiply:
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