GRE Math: Integers

GRE Integers is a part of GRE Arithmetic in the Quant section. Integers are the collection of whole numbers that can be positive {1, 2, 3, 4, …}, negative {-4, -3, -2, -1, …} or zero {0} but cannot be a fraction. Because a fraction is not a round number.

The word integer originated from the Latin word “Integer,” meaning whole or intact.

In Mathematics, integers are the collection of whole numbers (positive numbers) and negative numbers. The symbol of integers is Z. Here, the positive number is a real number that is greater than zero and the negative number is a real number that is smaller than zero. 

To understand it better, refer to the number line below.

Positive and Negative Numbers

Numbers that are on the right side of zero are called positive numbers and those to the left side are called negative numbers. The first ones must be written with the positive or plus (+) sign in front and the second ones with the negative or minus sign in front (-). These numbers are also identified as signed numbers.

Even and Odd Numbers
An even number is an integer that is “evenly divisible” by 2, i.e., divisible by 2 without a remainder.
An even number is an integer of the form n = 2k where k is an integer.

An odd number is an integer that is not evenly divisible by 2.
An odd number is an integer of the form n = 2k + 1, where k is an integer.

Zero is an even number.

ZERO:

1. 0 is an integer.
2. 0 is an even integer. An even number is an integer that is “evenly divisible” by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.
3. 0 is a neutral number and is neither a positive nor negative integer (the only one of this kind). 
4. 0 is divisible by EVERY integer except 0 itself.
5. The product of any number and a zero is zero. Basically, any number * 0 = 0.

Types of Integers

There are 6 types of integers. They’re as follows:

• Positive integers (e.g. 1, 3, 6, 7, 19, 32, etc.)
• Negative integers (e.g. -1, -3, -6, -7, -19, -32, etc.)
• Even integers (e.g. 2, 4, 8, 10, 16, 22, 26, -28, -26, -14, etc.)
• Odd integers (e.g. 5, 7, 27, 33, 53, etc.)
• Prime numbers are integers, divisible by only two positive integers (1 and the integer itself, e.g. 2, 3, 5, 7, etc).
• Coprime numbers are pairs of two positive integers which have 1 as an only common positive divisor. 

With Integers, we can perform all arithmetic operations like addition, subtraction, multiplication and division. 

1. Addition of two integers will be an integer (negative or positive).
2. Subtraction of two integers will be an integer (negative or positive).
3. Product of two integers will be an integer (negative or positive).
4. Division of two integers can be an integer or decimal (negative or positive).
5. Square of an integer (negative or positive) will be a positive integer.
6. Root of a positive integer can be an integer or decimal.  

Addition / Subtraction:

even +/- even = even;
even +/- odd = odd;
odd +/- odd = even.

Multiplication:

even * even = even
even * odd = even;
odd * odd = odd.

Multiplying Integers Rules

positive * positive = positive
positive * negative = negative
negative * negative = positive

Division:

Division of two integers can result in an even/odd integer or a fraction.
 
Dividing Integers Rules

positive / positive = positive
positive / negative = negative
negative / negative = positive

Examples:

1. 2 + 2 = 4
2. ( – 2) + ( – 2) = – 4
3. 2 – 1 = 1
4. ( – 2) – ( – 1) = – 1
5. 3 + 3 = 6
6. ( – 6) + ( – 6) = – 6
7. 4 – 2 = 2
8. ( – 4) – ( – 2) = – 2
9. 5 – 3 = 2
10. ( – 5) – ( – 3) = – 2
11. 3 * 3 = 9
12. ( – 3) * ( – 3) = 9
13. 4 * 4 = 16
14. ( – 4) * ( – 4) = 16
15. 4 * 5 = 20
16. ( – 4) * ( – 5) = 20
17. √4 = 2
18. √2 = 1.414…
19. 6/2 = 3
20. ( – 6) / ( – 2) = 3
21. ( – 6) / (2) = – 3
22. (6) / ( – 2) = – 3
23. 5/2 = 2.5
24. ( – 5) / ( – 2) = 2.5
25. (5) / ( – 2) = – 2.5
26. ( – 5) / (2) = – 2.5

Greatest Common Factor (Division) - GCF (GCD):

The Greatest Common Divisor (GCD) or the Greatest Common Factor (GCF) or Highest Common Factor (HCF) is the largest positive integer that divides the numbers without a remainder. 

You must do prime-factorization to find the GCF. Then, multiply the common factors (lowest power of the common factors).

Every common divisor of a and b is a divisor of gcd (a,b). 
a*b = gcd (a,b) * lcm (a,b)

Example: The greatest common divisor (GCD) of 15 and 30 is 15. This is because 15 is the largest number that divides both 15 and 30 without leaving a remainder.

Least Common Multiple (LCM):

The Lowest Common Multiple or Lowest Common Multiple (LCM) or Smallest Common Multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b. It can be divided by a and b without a remainder. In case if a or b is 0 (therein lies no positive integer), then LCM (a,b) is defined to be zero. 

You must do prime-factorization to find the LCM. Then multiply all the factors (highest power of the common factors). 

Example: The least common multiple (LCM) of 2 and 3 is 6. This is because 6 is the smallest number that is a multiple of both 2 and 3.

GRE Integers Practice Questions

1. If a is the greatest common divisor of 65 and 105 and b is the least common multiple of 28 and 16 then a + b = ?

1. 120
2. 117
3. 135
4. 200

Correct Answer: b) 117

Explanation: The greatest common divisor of 65 and 105 is 5 (represented as ‘a’), derived from the prime factorization. The least common multiple of 28 and 16 is 112 (represented as ‘b). When both are added, a + b, the answer is 117.
  
2. What is the least common multiple of 15 and 45?

1. 155
2. 40
3. 45
4. None of the above

Correct Answer: c) 45

Explanation: The least common multiple is the smallest number that is a multiple of all the numbers in the group. The list of some multiples of the two numbers and identify the smallest number in common to both.

Multiples of 15: 15, 30, 45, 60, 75, …
Multiples of 45: 45, 90, 135, 180, 225, 270, …

The smallest number in common is 45.

3. If x is a prime number, then 5x is

1. Odd
2. Even
3. Divisible by 2
4. Cannot be determined

Correct Answer: d) Cannot be determined

Explanation: A prime number like 2 or 5 and multiple with 5.

It can be 10 or 25, which is odd and even, so that we can’t conclude that 5x is either even or odd. It is also not divisible by 2. So the answer is “Cannot be determined.”

4. Quantitative Comparison

Quantity A: The number of prime numbers between 0 and 100, inclusive.
Quantity B: The number of prime numbers between 101 and 200, inclusive.
Quantity C: The number of prime numbers between 201 and 300, inclusive.

1. Quantity A is greater
2. Quantity B is greater
3. Quantity C is greater
4. The relationship cannot be determined from the information given

Correct Answer: a) Quantity A is greater

Explanation: In the number line, the number of primes between 0 and 100 is far more than between 101 and 300. Therefore, Quantity A is greater than Quantity B and C.

5. A prime number is divisible by:

1. It depends on the prime number
2. 0
3. 1
4. Two answers are correct

Correct Answer: d) Two answers are correct.

Explanation: A prime number is a number divisible by only one and itself. A prime number is not divisible by zero, because numbers divided by zero are undefined.

6. Which of the following integers are factors of both 125 and 70?

1. 7
2. 5
3. 3
4. 12

Correct Answer: b) 5

Explanation: 5 is the only answer that is a factor of both 125 and 70. 125/5 = 25 and 70/5 = 14. The other answers are either a factor of 125 or 70 or neither, but not both.

7. The product of two distinct integers is — 187. Which of the following is a possible sum of these two integers?

1. — 6
2. — 18
3. — 188
4. — 28

Correct Answer: a) — 6

Explanation: — 187 is actually the product of a positive and negative integer. Consider all of the ways that it could be factored, and the sums these factors would produce: 

— 1, 187; — 1 + 187 = 186
1, — 187; 1 + ( — 187) = — 186
— 11, 17; — 11 + 17 = 6
11, — 17; 11 + ( — 17) = — 6

— 6 is the answer choice that matches.