# GRE® Number Properties

#### Concepts Tested in Number Properties, Number Sytems, and Number Theory

One can expect three to five questions from number properties, number system, and number theory in the quant section of the GRE General Test. The concepts are core concepts and you need to get an in depth understanding of these concepts to ace these questions in the GRE quant section. For instance, it is not only sufficient that you know how to find the LCM and HCF of numbers, but you also need to know what is the idea behind finding LCM and in what cases do we find LCM. The following concepts are tested in Number Theory in the GRE test.

• Operations on numbers. Results of operations on odd and even numbers.
• Prime numbers - how to determine whether a number is prime. Composite numbers. Co prime numbers. Prime factorizing a number.
• Properties of squares of numbers, cubes of numbers.
• LCM and HCF of numbers. When to find the LCM or HCF. LCM and HCF of fractions.
• Remainders, remainders of division of the product and sum of numbers by the same divisor.
• Factorials, number of trailing zeroes.
• Finding units digit of numbers. Finding tens digit of numbers

1. N is the smallest positive integer that has 7 factors.

Column A Column B
Number of factors of √N Number of factors of (N - 2)
1. Column A is greater
2. Column B is greater
3. The two columns are equal
4. Cannot be determined
Approach to solve this GRE Number Properties QC question

Step 1: Find the interger N
Step 2: Evaluate Column A. Compute the number of factors of √N
Step 3: Evaluate Column B. Compute the number of factors of (N - 2)
Step 4: Compare values in step 2 and step 3 to find the answer to this GRE quantitative comparison question

2. How many pairs of natural numbers exist whose HCF is 12 and add up to 216?

1. 6
2. 3
3. 9
4. 18
5. 17
Approach to solve this Number Properties question

Step 1: The two numbers can be written in the form 12x + 12y = 216, where x and y are positive integers and coprime to each other
Step 2: So, x + y = 18. List possible values of x and y that are co prime to each other and add up to 18

3. Which of the following numbers are factors common to 108, 288, and 396?
Indicate all such numbers.

1. 72
2. 24
3. 36
4. 8
5. 12
6. 9
Approach to solve this Number Properties question

Step 1: Find the HCF of the three numbers.
Step 2: List down all the factors of the HCF.
Step 3: Select from the given answer options, numbers that are factors of the HCF.

4. When a positive integer x is divided by 12, the remainder is 5.

Quantity A Quantity B
Remainder when 2x is divided by 12. Remainder when 3x is divided by 12.
1. Quantity A is greater
2. Quantity B is greater
3. The two Quantities are equal
4. Cannot be determined
Approach to solve this GRE Number Properties QC question

Step 1: Express x as 12q + 5 (Euclid's Division Lemma)
Step 2: Quantity A: Compute the remainder when 2x, i.e., 24q + 10 is divided by 12
Step 3: Quantity B. Compute the remainder when 3x, i.e., 36q + 15 is divided by 12
Step 4: Compare values in step 2 and step 3 to find the answer to this GRE quantitative comparison question

5. Quantitative Comparison

Quantity A Quantity B
$$sqrt{\frac{143}{208}}$ × $\sqrt{\frac{169}{77}}$ $\sqrt{\frac{459}{306}}$ × $\sqrt{\frac{224}{675}}$ 1. Quantity A is greater 2. Quantity B is greater 3. The two Quantities are equal 4. Cannot be determined Approach to solve this GRE Number Properties QC question Step 1: Prime factorize the numerator and denominator of the terms in both Quantity A and Quantity B. Step 2: Quantity A. Simplify the term by bringing the fraction to its least form. Take perfect squares out of the root to further simplify the expression. Step 3: Quantity B. Simplify the term by bringing the fraction to its least form. Take perfect squares out of the root to further simplify the expression. Step 4: Compare values in step 2 and step 3 to find the answer to this GRE quantitative comparison question. 6. If n is a positive integer that leaves a remainder of 45 when divided by 60, which of the following cannot be a divisor of n? 1. 9 2. 15 3. 45 4. 30 5. 25 Approach to solve this Number Properties question Step 1: Express 'n' using Euclid's division algorithm as 60q + 45 Step 2: 60q is an even number. So, 60q + 45 is an odd number. Step 3: Odd numbers are not divisible by even numbers. Identify options to arrive at answer. 7. Which of the following numbers has an even number of positive integer factors? Indicate all such numbers? 1. 29 2. 116 3. 216 4. 384 5. 676 6. 576 Number Properties Select One or More Answers Question Step 1: What kind of numbers have an odd number of factors? Perfect Squares. Step 2: Identify from answer options numbers that are not perfect squares. 8. Which of the following numbers will divide 5040 completely without leaving a remainder? Indicate all such numbers? 1. 243 2. 144 3. 240 4. 112 5. 560 6. 405 Approach to solve this Number Properties question GRE Method: From the GRE exam point of view, use the on screen calculator the pick the options one at a time and check whether the division yields an answer that is an integer. For the Curious: A number properties approach to solve the question is given in the video and the answer explanation webpage. 9. x, y, and z are distinct integers such that |x|, |y|, and |z| ≤ 5. What is the least possible value of xyz? 1. 0 2. -125 3. -60 4. -100 5. -75 Approach to solve this Absolute Values Question Step 1: We can infer that -5 ≤ x, y, z ≤ 5 Step 2: The value is least when the product is negative. In negative numbers, greater the magnitude, lesser the value. Step 3: Try to maximize magnitude keeping in mind that the numbers are distinct integers. 10. Numeric Entry | If$3-22 + 3-19 + 3-17 + 3-16) = k(3-24), what is k?

Approach to solve this Number Properties question

Step 1: Express each of the terms 3-22, 3-19, 3-17, and 3-16 in terms of (3-24)
Step 2: Take (3-24) common and add the remaining numbers to compute the value of k

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