Factors And Multiples- 2 - Free MCQ Practice Test with solutions, GMAT

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.

To determine which of the options cannot be the greatest common divisor (GCD) of two different prime numbers a and b, we need to understand the properties of prime numbers.

Prime numbers have only two distinct positive divisors: 1 and themselves. Thus, the GCD of any two different prime numbers is always 1, since they do not share any factors other than 1.

• a - b: This represents the difference between the two primes and can be greater than 1, but it cannot be a divisor common to both.
• b - a: Similar to the previous point, this is just the negative of the first option and does not affect the GCD.
• 2a - b: This also does not have to equal 1 and can be any number depending on the values of a and b.
• a b: This represents the product of a and b, which is much larger than 1, and cannot be a GCD as it is not a common divisor.

Since the GCD of two different prime numbers is always 1, the only option that cannot be the GCD is a b, as it is greater than 1.

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.

The least common multiple (LCM) of two distinct positive integers a and b cannot be a + b.
Instead, it is the smallest number that is a multiple of both a and b, which is either a or b if one divides the other.

Hence , Option D is correct.

To find a unique integer x, we need both pieces of information together:

Statement 1 alone (lcm(x,16)=48) ⇒ x could be 3, 6, 12, 24, or 48 (all give LCM = 48).
Statement 2 alone (gcd(x,16)=4) ⇒ x could be 4, 12, 20, 28, … infinitely many.

Only by using both do we pin down x:

• From lcm=48 we get x=3·2ᵏ with k≤4 ⇒ {3,6,12,24,48}.

• From gcd=4 we need x divisible by 4 but not by 8 ⇒ leaves x=12.

Hence both statements are required.

Answer: (b) Both statements are required to answer the question.

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.