If a and b are positive integers and k is the greatest common factor

Question

If a and b are positive integers and k is the greatest common factor of a and b, then k must be the greatest common factor of a and which of the following integers?

A. a + b
B. 3 + b
C. ab
D. 3b
E. b^2

SHoylandBPrep · Answer

One way to do this problem is to plug in for a and b. There are many numbers you could use. Let's use  a = 4  and  b = 2 . Then the GCF is  k = 2 . Now look at the answer choices:

→ A) a + b = 6. The GCD of 4 and 6 is 2.  KEEP. 
→ B) 3 + b = 5. The GCD of 4 and 5 is 1.  ELIMINATE. 
→ C) ab = 8. The GCD of 4 and 8 is 4.  ELIMINATE. 
→ D) 3b = 6. The GCD of 4 and 6 is 2.  KEEP. 
→ E) b^2 = 4. The GCD of 4 and 4 is 4.  ELIMINATE.

Only A and D are left. It's probably safe to eliminate D because the 3 seems pretty random. If you are unsure, you could also plug in again to decide between the two. Try to find numbers that will eliminate D since it is suspect. You could make b one-third of a.

If  a = 9  and  b = 3 , then the GCF is  k = 3 .

→ A) a + b = 12. The GCD of 9 and 12 is 3.  KEEP. 
→ D) 3b = 9. The GCD of 9 and 9 is 9.  ELIMINATE.

The answer is  (A) .

nick1816 · Answer

a=k*m
b=k*n

where m and n are coprimes

if m is coprime to n, then m or n is coprime to both m+n and m-n.

a+b=k(m+n) and a=k*m; GCD(a+b,a)=k

A

ScottTargetTestPrep · Answer

Solution:

We can let a = 4 and b = 6. So, we have k = GCF(4, 6) = 2. Now let’s check the given answer choices.

A) a + b = 4 + 6 = 10 and GCF(4, 10) = 2 → This could be the correct answer.

B) 3 + b = 3 + 6 = 9 and GCF(4, 9) = 1 → This is not the correct answer.

C) ab = 4(6) = 24 and GCF(4, 24) = 4 → This is not the correct answer.

D) 3b = 3(6) = 18 and GCF(4, 18) = 2 → This could be the correct answer.

E) b^2 = 6^2 = 36 and GCF(4, 36) = 4 → This is not the correct answer.

We see that we can eliminate choices B, C and E, but we still have choices A and D being the possible correct answer. We need to determine which of these two is the correct answer. So let’s now let a = 6 and b = 4, and we see that k = GCF(6, 4) = 2.

A) a + b = 6 + 4 = 10 and GCF(6, 10) = 2 → This could be the correct answer.

D) 3b = 3(4) = 12 and GCF(6, 12) = 6 → This is not the correct answer.

Answer: A

Kinshook · Answer

If a and b are positive integers and k is the greatest common factor of a and b, then k must be the greatest common factor of a and which of the following integers?

a = kx; where a & x are co-prime positive integers
b = ky; where a & y are co-prime positive integers

a+b = k(x+y)

k is greatest common factor of a & a + b.

IMO A

If a and b are positive integers and k is the greatest common factor

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If a and b are positive integers and k is the greatest common factor

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