Test: Prime Numbers- 1

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

Correct Answer :- a

Explanation : The answer should be A. 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.

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.

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.

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

y² 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.

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.

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

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

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).

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

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

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.