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QUESTION: 1

If *m *and *n *are two different prime numbers, then the least common multiple of the two numbers must equal which one of the following?

Solution:

One of the great things about Integer Properties questions if that we can often solve them by finding values that satisfy the given condition.

And this question to given condition is: m and n are two different prime numbers

So, it COULD be the case that m = 2 and n = 3

If m = 2 and n = 3, then the least common multiple of m and n is 6 (since 6 is the least common multiple of 2 and 3)

Now we can plug m = 2 and n = 3 into each answer choice to see which one yields an output of 6

A) mn = (2)(3) = 6. KEEP!

B) m + n = 2 + 3 = 5. No good. We want an output of 6.

C) m - n = 2 - 3 = -1. No good. We want an output of 6.

D) m + mn = 2 + (2)(3) = 8. No good. We want an output of 6.

By the process of elimination, the correct answer must be A

QUESTION: 2

Is the product of two numbers *x *and *y *a prime number?

1) *x *and *y *are two prime numbers.

2) *x *and *y *are two odd numbers not equal to 1.

Solution:

**Correct Answer :- a**

**Explanation **: The answer should be D. Only the possibility of 2 different numbers multiplying and creating a prime number is when one number is 1 and another is prime itself. Both the statements clear that situation.

QUESTION: 3

How many prime numbers exist between 200 and 220?

Solution:

Odd numbers between 200 and 220 are:

201, 207, 210, 213, 219 are divisible by 3 (because the sum of their digits is divisible by 3).

205, 215 are divisible by 5.

Hence, we have to check just the following numbers: 203, 209, 211 and 217. Now,

203 = 7*29 (Not prime).

209 = 11*19 (Not prime).

211 = Prime

217 = 7*31 (Prime).

So there is only one prime number between 200 and 220.

QUESTION: 4

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

Solution:

If x = 2 and y = 3 then the sum of these prime numbers is 5..

If x = 2 and y = 7, then sum of these prime numbers 9

If x = 3 and y = 13, then then sum of these prime numbers is 16

But in case of option D there are no two prime numbers that can be added to get 23.

Hence option D is correct.

QUESTION: 5

An integer greater than 1 that is not prime is called composite. If the two-digit integer *n* is greater than 20, is *n* composite?

1) The tens digit of *n* is a factor of the units digit of *n*

2) The tens digit of *n* is 2

Solution:

QUESTION: 6

If *x* is a positive integer, is *x* prime?

1) *x* has the same number of factors as *y2*, where *y* is a positive integer greater than 2.

2) *x* has the same number of factors as *z*, where *z* is a positive integer greater than 2.

Solution:

(1) x has the same number of factors as y^{2}, where y is a positive integer greater than 2.

y^{2} is a perfect square. The number of distinct factors of a positive perfect square is ALWAYS ODD, while the number of factors of a prime is two (1 and itself). Thus since x has the same number of factors as a perfect square it cannot be a prime. Sufficient.

(2) x has the same number of factors as z, where z is a positive integer greater than 2. Clearly insufficient.

QUESTION: 7

Which of the following could be the median of a set consisting of 6 different primes?

Solution:

QUESTION: 8

Set A consists of 8 distinct prime numbers. If *x* is equal to the range of set A and *y* is equal to the median of set A, is the product *xy* even? ?

1) The smallest integer in the set is 5.

2) The largest integer in the set is 101. ?

Solution:

QUESTION: 9

If x is an integer, is x! + (x + 1) a prime number?

1) x < 10

2) x is even

Solution:

QUESTION: 10

If *k* is a positive integer. Is *k* a prime number??

1) No integers between "2" and "square root of *k*" inclusive divides *k* evenly

2) No integers between 2 and *k*/2 divides *k* evenly, and *k* is greater than 5.

Solution:

QUESTION: 11

Is the product of three integers *xyz* a prime number?

1) *x = -y*

2) z* = 1 *

Solution:

**Correct Answer :- a**

**Explanation :** xyz will be prime only when x = 1, y = -1, z = -2

or x = -1, y = 1, z = -2

(Sufficient)

if z=1, no other possible values of x and y make a prime product.

QUESTION: 12

If p is a prime number greater than 2, what is the value of p?

1) There are a total of 100 prime numbers between 1 and p + 1

2) There are a total of p prime numbers between 1 and 3912

Solution:

QUESTION: 13

Is the product of two numbers *x *and *y *a prime number?

1) x + y = prime

2) y is not prime

Solution:

**Correct Answer :- d**

**Explanation :** Given: x,y are integers > 0.

is x*y = prime?

prime number = 1*prime.

statement 1:

x = prime - y

possible values of x,y:

(3,1): product is a prime

(4,1):product is not a prime

not sufficient

statement 2:

y ≠≠ prime

nothing is specified about x.

not sufficient

combining both statements,

possible values of x,y:

(3,1): product is a prime

(4,1):product is not a prime

QUESTION: 14

Is the product of two numbers *x *and *y *a prime number?

1) x - y = prime

2) y is not prime

Solution:

**Correct Answer :- d**

**Explanation :** Given: x,y are integers > 0.

is x*y = prime?

prime number = 1*prime.

statement 1:

x = prime + y

possible values of x,y:

(3,1): product is a prime

(4,1):product is not a prime

not sufficient

statement 2:

y ≠≠ prime

nothing is specified about x.

not sufficient

combining both statements,

possible values of x,y:

(3,1): product is a prime

(4,1):product is not a prime

QUESTION: 15

Is the product of two numbers *x *and *y *a prime number?

1) x/y = prime

2) x and y are consecutive integers

Solution:

**Correct Answer :- b**

**Explanation : **Both the statements are required to answer the question.

Because it is given x and y are consecutive integers

Let x = 2, y = 1

When dividing x/y = 2/1

= 2(prime).

QUESTION: 16

Is the product of two numbers *x *and *y *a prime number?

1) x is even

2) y is odd

Solution:

**Correct Answer :- d**

**Explanation : **Given: x,y are integers > 0.

is x*y = prime?

prime number = 1*prime.

statement 1:

x is even

possible values of x,y:

(1,2): product is a prime

(1,4):product is not a prime

not sufficient

statement 2:

y is odd

nothing is specified about x.

not sufficient

combining both statements,

possible values of x,y:

(3,1): product is a prime

(4,1):product is not a prime

QUESTION: 17

Is the product of two numbers *x *and *y *a prime number?

1) x + y = even

2) x is even

Solution:

**Correct Answer :- B**

**Explanation :** 1) x + y = even

If x=2, y=2, xy=4 which is not prime

If x=1, y=3, xy=3 which is prime

1 is not sufficient

(2) x is even

If x=2, y=2, xy=4 which is not prime

If x=2, y=1, xy=2 which is prime

2 is not sufficient

(1)+(2)

x+y=even and x=even

So y=even

xy=even*even which must be divisible by 4 and so xy is not prime

(1)+(2) is sufficient

QUESTION: 18

If *k* is a positive integer, is *k* a prime number??

1) *k* can be written as 6*n* + 1, where *n* is a positive integer.

2) k > 4!

Solution:

QUESTION: 19

If *k* is a positive integer, is *k* a prime number?

1) *k* is the sum of three consecutive prime numbers

2) *k* has only 2 factors

Solution:

Statement 1: k is the sum of three consecutive prime numbers

2 + 3 + 5 = 10, which is not a prime number.

11 + 13 + 17 = 41, which is a prime number.

Therefore Statement 1 Alone is Insufficient. Answer options could be B, C or E

Statement 2: k has only 2 positive factors

If k has only 2 positive factors, then they have to be 1 and k itself, which is the definition of a prime number. k is a prime number.

Therefore Statement 2 Alone is Sufficient.

QUESTION: 20

Is the integer *x* a prime number?

1) * x + 1 * is prime

2) x* + 2 *is not prime

Solution:

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