Test: Factors And Multiples- 2 - GRE MCQ

The greatest common divisor (GCD) of two positive integers a and b is the largest positive integer that divides both a and b without leaving a remainder.

Out of the given options, the only number that cannot be the GCD of two positive integers a and b is option 4, a - 2b.

To see why, consider the following example: let a = 10 and b = 4. The factors of 10 are 1, 2, 5, and 10, and the factors of 4 are 1, 2, and 4. The common factors of 10 and 4 are 1 and 2, and the greatest common factor is 2.

Now, if we substitute a = 10 and b = 4 into the expression a - 2b, we get:

a - 2b = 10 - 2(4) = 2

Since 2 is the GCD of 10 and 4, it is possible for a - 2b to be the GCD of two positive integers. However, there are other cases where a - 2b is not the GCD of two positive integers, so it cannot be concluded that a - 2b is always a valid option for the GCD of two positive integers.

Therefore, the correct answer is option 4, a - 2b.

Understanding the GCD of Two Different Primes

• Definition of Primes: Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves.
• GCD of Two Different Primes: Since a and b are different primes, their only common positive divisor is 1. Therefore, the GCD of a and b is 1.

Evaluating Each Option

Option a) a - b

• Possible Value: a - b can be 1 when a and b are consecutive primes.
• Example: Let a = 3 and b = 2.
  • a - b = 1
• Conclusion: a - b can be 1, so it can be the GCD of a and b.

Option b) b - a

• Possible Value: b - a can be 1 when b and a are consecutive primes.
• Example: Let a = 2 and b = 3.
  • b - a = 1
• Conclusion: b - a can be 1, so it can be the GCD of a and b.

Option c) 2a - b

• Possible Value: 2a - b can be 1 depending on the values of a and b.
• Example: Let a = 2 and b = 3.
  • 2a - b = 4 - 3 = 1
• Conclusion: 2a - b can be 1, so it can be the GCD of a and b.

Option d) a + b

• Possible Value: Since a and b are primes greater than 1, a + b will be at least 2 + 3 = 5.
• GCD Analysis: The GCD of a and b is 1, and the GCD cannot be larger than the smallest of the two numbers.
• Conclusion: a + b cannot be 1 and cannot be the GCD of a and b.

Answer

Option d) a + b cannot be the greatest common divisor of two different prime numbers a and b.

Option D: a + b

t means any two prime numbers will have only one common factor and that would be '1', as per the definitions of prime number and highest common factor. Hence, any two different prime numbers will have the highest common factor as '1'. It means the H.C.F. of given two prime numbers a and b is 1.

375 = (3)(5)(5)(5) = (3)(5)(5²)
In order for 375y to be a perfect square, the prime factorization of y must contain at least one 3 and one 5.
In other words, y must be a multiple of 15.

If y is a multiple of 15, then y/15 must be an integer and y²/25 must be an integer.