Prime Numbers- 1 - Free MCQ Practice Test with solutions, GMAT Quant Reasoning

• One of the great things about Integer Properties questions is that we can often solve them by finding values that satisfy the given condition.
• And this question to the 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

• Prime numbers that lie between 70 and 100 are 71, 73, 79, 83, 89, 97
• The sum of all the terms = 71 + 73 + 79 + 83 + 89 + 97
• ⇒ The sum of all the terms = 4 92
• Total number of terms = 6
• ⇒ Average = 492/6
• ⇒ Average = 82
• Hence, the correct answer is 82.

• 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.

• The number which is divisible by 1 and itself ,is known as the prime number.
• 233 is divisible by 1. 
• 253 is divisible by 11. 
• 247 is divisible by 13 
• Hence, only 233 is the number which is not divisible by any number except 1 and itself.

• Statement 1 : Not Sufficient
• 2! + (2+1) =2x1 + 3 = 5 (prime number)
• 3! + (3+1) = 3x2x1 + 4 = 10 (not a prime number)
• Statement 2 : Not Sufficient
• 2! + (2+1) =2x1 + 3 = 5 (prime number)
• 4! + (4+1) = 4x3x2x1+5 = 29 (prime number)
• 6! + (6+1) = 6x5x4x3x2x1+7 = 727 (prime number)
• 8! + (8+1) = 8x7x6x5x4x3x2x1 9 = 40329 (not a prime number)
• Combined Statements 1 and 2 are not sufficient because when x<10 and x is even you get prime and non-prime numbers.

• Statement 1:
• Property of a prime number: k is a prime number if no integer between 2 and square root of k (inclusive) is a factor of k.
• Sufficient
• Statement 2:
• k/2 >= sqrt(k)
• So it satisfies the above-mentioned property.
• Sufficient
   

Solution and Explanation
Approach Solution 1:
The situation given is that p is a prime number greater than 2 and it is required to determine the value of p.

However, it is important to focus on the part that although it has been asked for the value of p, it is actually needed to determine whether the value of p is unique. Accordingly, from the given two conditions or statements in relevance to the question, it is to be determined that the value of p is unique. If the same is determined then the given statements would be considered sufficient. Any one of the statements or even both the statements can be sufficient or insufficient. Hence, it is important to judge and analyze both statements individually.

Statement One Alone:

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

If there are exactly 100 prime numbers between 1 and p + 1, then there are exactly 100 prime numbers in the list. This includes- 2, 3, 5, 7, 11, 13, …, p. Whatever value p is, p must be unique. 

Accordingly, even if it is the 100th number in the list, p would be unique. Hence, Statement one alone can be considered sufficient in determining the value of p. Considering this aspect, the options, a, b and d can be eliminated.

Statement Two Alone:

The second statement states that- There are a total of p prime numbers between 1 and 3,912.

It is a fact that between two distinct positive integers, there must be a unique number of primes. For example, between 1 and 10 inclusive there are exactly 4 primes: 2, 3, 5, 7. There can’t be 3 primes or 5 primes between 1 and 10. 

Therefore, the same logical explanation is applied to the second statement in order to find the value of p. Accordingly, it can be identified that between 1 and 3,912, there must be a number of prime numbers. These prime numbers must definitely be unique as well. Although the actual value of p cannot be determined from this statement, it can be considered that p must be unique. Hence, the second statement can also be considered to be absolutely sufficient as a separate and alone statement. 

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
• Not sufficient.

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
• 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 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).

• Given: x and y are integers greater than 0.
• For x*y to be prime, one of the factors must be 1 and the other must be a prime number since a prime number has exactly two distinct positive factors: 1 and itself.
• Statement 1: x is even. For instance, if x equals 2 (the only even prime), then for x*y to be prime, y must be 1. However, x being even alone does not ensure that y is 1.
• Statement 2: y is odd. This provides no restriction on x, so the statement by itself is insufficient.
• Combining both statements: With x being even, a valid choice is x = 2. Then possible combinations include (2, 1) where the product is 2 (a prime) and (2, 3) where the product is 6 (not a prime). Hence, even when both conditions are considered together, the information is insufficient to determine uniquely whether the product is prime.

• If x+y is even, it doesn't provide any information about whether xy is prime or not. For instance, 2+4=6 is even, and 2×4=8 is not a prime number, but 2+6=8 is even, and 2×6=12 is also not prime. So, statement 1 alone is insufficient.

• If x is even, it also doesn't provide any information about whether xy is prime or not. For example, if x=2, y can be any number, and the product xy might or might not be prime. So, statement 2 alone is insufficient.

• Combining the two statements doesn't provide any new information; it doesn't help determine whether the product of x and y is prime.
• Therefore, the correct answer is:
• More information is required as the information provided is insufficient to answer the question

• Statement 1:k can be written as 6n + 1, where n is a positive integer.
• When n = 1: 6n + 1 = 7 which is a prime number
• When n = 4: 6n + 1 = 25 which is not a prime number
• Therefore Statement 1 Alone is Insufficient. Answer options could be B, C or E
• Statement 2: k > 4! or k > 24
• Here we can have multiple values. If k = 25, it is not prime and if k = 29 it is prime.
• Therefore Statement 2 Alone is Insufficient.
• Combining Both Statements:
• k = 6n + 1 and k > 24
• When n = 4: k = 6n + 1 = 25 which is not a prime number
• When n = 5: k = 6n + 1 = 31 which is a prime number
• Therefore Both Statements together are Insufficient.

Statement 1 (k is the sum of three consecutive primes) is not sufficient.

• Example 1: 2 + 3 + 5 = 10, which is not prime.

• Example 2: 5 + 7 + 11 = 23, which is prime.
  Since some sums give composites and others give primes, knowing that k is the sum of three consecutive primes doesn’t guarantee k itself is prime.

5Statement 2 (k has only 2 positive factors) is both necessary and sufficient.

• By definition, a positive integer with exactly two positive divisors (1 and itself) is prime.

• Thus statement 2 alone answers the question.

• If k has only 2 factors, we immediately know k is prime; if k had more than 2 factors, it wouldn’t be prime.

Explanation:

• Statement 1 alone is not sufficient to determine if x is a prime number. For example, if x = 2, then x + 1 = 3 is prime, but if x = 4, then x + 1 = 5 is also prime.


• Statement 2 alone is also not sufficient to determine if x is a prime number. For example, if x = 3, then x + 2 = 5 is prime, but if x = 4, then x + 2 = 6 is not prime.


• Combining both statements does not provide enough information to determine if x is a prime number. For example, if x = 2, then x + 1 = 3 is prime and x + 2 = 4 is not prime. However, if x = 4, then x + 1 = 5 is prime and x + 2 = 6 is not prime.


• Therefore, more information is required to determine if the integer x is a prime number.