Number Theory

Question 1

If m = 44n/ 3,630 and m and n are integers, then what is the least possible value of n?

55 
75
165
330
825

Answer 1

Answer = 165

Prime factorization of 44n and 3,630 help here

Question 2

If the remainder when positive integer y is divided by 24 is 14, then which of the following must be a factor of y?

31
19
7
3
2

Answer 2

So if you say 24 goes into it once and then you add 14 more you have 38
So what are the factors of 38? = 1 * 38 and 2 * 19
Then you can try another number so what is 24 goes into the number twice and then add 14 so = 62 now what are the factors of 62 = 1 * 62 and 2 * 31 so since 2 is going into both questions, answer = 2

Question 3

When a GRE question involves factors or multiples, what should you do?

Answer 3

Rewrite the numbers in the problem as the product of their primes

Question 4

The remainder when 5 is divided by 9

Answer 4

9 divides into 5 zero times, so 5 is left over.

answer = 5

Question 5

The remainder when 11 is divided by 3 is y. If 24 is divided by y, the remainder is?

Answer 5

11 divided by 3 is 3 with 2 left over, so y = 2. If 24 is divided by y (which y = 2) it divides evenly and there is no remainder so the answer = 0.

Question 6

The remainder when 29 is divided by x is 1, and the remainder when 66 is divided by x is 3, so x = ?

Answer 6

Subtract the remainder from 29 and then consider the factors of the resulting number. So 29 - 1 = 28. The factors of 28 are 1, 2, 4, 7, 14 and 28.

Now do the same for 66. So, 66 - 3 = 63. The factors of 63 are 1, 3, 7, 9, 21, and 63.

The only number that is common in both and meets the criteria of the question is 7. So, the answer = 7

Question 7

The remainder when positive integer x is divided by 6 is 4. If x > 4, then the value of x is ?

Answer 7

Find the value of x. An easy way to do that is to add the divisor (6) and the remainder (4) so 6 + 4 = 10 and x could = 10, which is one of the options. Since x is really 4 more than a multiple of 6, other multiples of 6 could also work. So, x could also equal 12 +4 = 16 or 18 + 4 = 22 (using different multiples of 6) However, since the question asks for the least value of x, the correct answer = 10.

Question 8

Remainders are the most common number theory questions on the GRE !!!

Question 9

If the remainder when integer x is divided by 4 is 3 and the remainder when integer y is divided by 4 is 2, what is the remainder when x + y is divided by 4?

Answer 9

x/4 Remainder 3, so x = 4 + 3 = (7)

y/4 Remainder: 2, so y = 4 + 2 = (6)

x + y = 13

(x + y) / 4: Remainder = 1

an easy way to calculate x and y is to add the remainder to the divisor so in this case the divisor is 4 and then you add each remainder to it.

Question 10

7 / 3 = 2 remainder 1

in words “seven divided by three”
is asking?

Answer 10

how many times does 3 go into 7? The remainder is the whole number that is left over so the answer is = 2 remainder 1

Question 11

fifteen divided by four ?

Answer 11

15 / 4 = 3 remainder 4

How many groups of 4 can you fit into 15? Then what is the whole number that is left over

Question 12

25 / 7 = ?

Answer 12

= 3 with remainder 4

Question 13

30 / 7 = ?

Answer 13

7 * 4 = 28

so answer = 4 remainder 2

Question 14

41 / 6 = ?

Answer 14

Answer = 6 remainder 5

Question 15

3771 is divided by 8 what is your remainder?

Answer 15

what you are dividing goes under the shape and what you are dividing by goes outside the shape

Question 16

the number x “is divided by” y, which number goes on the outside?

Answer 16

y goes on the outside

Question 17

If x and y are positive integers and 15x = sqrt(735y), what is the least possible value of x + y?

Answer 17

15x = sqrt(735y)

Start by re writing the numbers in the problem as the product of their prime factors. So, 15 = 3 * 5 and 735 = 3 * 5 * 7 * 7 and then rewrite the equation as (3 * 5)x = sqrt(( 3 * 5 * 7 * 7)y)
Next, because the question says that x and y are positive integers, in order for the square root of a number to be an integer, there must be an even number of each number under the radical sign. There are already two 7’s and since 7^2 = 49 and the sqrt(49) = 7, the radical can be rewritten, (3 * 5)x = 7sqrt((3 * 5)y) and then you know x = 7.
So, (3 * 5) = sqrt((3 * 5)y)
To solve for y, square both sides of the equation.
You get, 3^2 * 5^2 = 3 * 5y Then divide each side of the equation by 3*5 and you get y = 3 * 5
x = 7
y = 15
x + y = 22

Question 18

In order for the square root of a number to be an integer, there must be an even number of each number under the radical sign

Question 19

Even integers, when divided by 2 produce a remainder of what?

Answer 19

zero

Question 20

Odd integers, when divided by 2 produce a remainder of what?

Answer 20

1

Question 21

Which of the following numbers are both multiples of 3 and factors of 81?

3
6
9
18
27
81

Answer 21

To find if they are factors of 81, what you can do is prime factor 81 and also prime factor each of the answer options.

The prime factorization of 81 is (3 * 3 * 3 * 3)

prime factor or 3 is 3 - is there at least one 3 in the prime factors of 81? yes

prime factors of 6 is 3 * 2 is there a 3 in the pf of 81? yes , but is there a 2, no so 6 is not a factor of 81

prime factors of 9 yes, there are at least two threes in 81

prime factors of 18 are 2 * 3 * 3, in 81 there are two 3’s but no 2 so 18 is not a factor of 81

prime factor of 27 is 3^3 so yes

and 81 yes

Question 22

Take any number that the divisor divides into evenly

So if you are looking for something that when divided by 4 gives you a remainder of 3 for example think about 8 + 3 = 11
11/ 4 = 2 remainder 3

remainders are the numerators in the fraction part of a mixed number