If a and b are positive integers, which of the following CANNOT be the

Question

If a and b are positive integers, which of the following CANNOT be the greatest common divisor of 15a and 40b?

(A) 5
(B) 4a
(C) 5(a - b)
(D) 40b
(E) 15a

Mokshagna123 · Answer

The question is typed wrong.
The actual question is as follows:
If a and b are positive integers, which of the following CANNOT be the greatest common divisor of 15a and 40b?

and the answer to it is option B - 4a

purplex · Answer

Is it B because none of the numbers have 4 as a common factor or am I missing something here?

chetan2u · Answer

The numbers are 15a and 40b, where a and b can take value of any positive integers.

We can generalise. 
 A factor that is not there in constant term multiplied by the variable of that term.  
So, 7a, 11a, 100a etc will not divide 15a, while 3b, 7b, 11b, 100b will not divide 40b.

4a is the answer as it will not divide 15a.

Also other way would be if the divisor is not the greatest.  
For example: 20a and 40b will not have GCD as 5 or 10 but 20a or 20b or 40b or 14520 or any other number that contains 20 as a factor.

But let us check how other options fit in. 
(A) 5……. Let a=7 and b=11, then both 15a and 40b are divisible by 5.  
(B) 4a…… 4 is not a factor of 15, so 4a will not be a factor of 15a. 
(C) 5(a - b)…. a=2 and b=1, will make it 5 
(D) 40b…… if a=40b, then 15a and 40b have the common divisor 40b 
(E) 15a……  if b=15a, then 15a and 40b have the common divisor 15a

B

Addu.23 · Answer

I have major confusion with the explanation of D & E options. If A/B can be assumed then in option B 'a' in 4a can be assumed as 40. That would negate B option too right?

Bunuel

Bunuel · Answer

No. If 4a = 40, then a = 10, meaning 15a = 60 and 4a = 40. Since 40 is not a divisor of 60, it cannot be the greatest common divisor of 15a and 40b. If a and b are positive integers, which of the following CANNOT be the greatest common divisor of 15a and 40b? (A) 5 (B) 4a (C) 5(a - b) (D) 40b (E) 15a The greatest common divisor (GCD) of 15a and 40b must, of course, divide both 15a and 40b. Glancing at the options, we can quickly rule out option B because it isn't a divisor of 15a. This is clear since 15a/4a = 15/4, which is not an integer. Since there can only be one correct answer, B must be it. Now, if we want to verify the other options: For 15a = 3 * 5 * a and 40b = 2^3 * 5 * b , the GCD is found by taking the common prime factors in their lowest powers between the two expressions. (A) 5 If a = b = 1, then 15a = 15 and 40b = 40. The GCD of 15 and 40 is indeed 5. (C) 5(a - b) If a = 3 and b = 2, then 15a = 45 and 40b = 80. The GCD of 45 and 80 is 5(a - b) = 5. (D) 40b If a = 40b, then 15a = 3 * 5 * a = 2^3 * 3 * 5^2 * b. The GCD of 2^3 * 3 * 5^2 * b and 2^3 * 5 * b, is 2^3 * 5 * b = 40b. (E) 15a If b = 15a, then 40b = 2^3 * 5 * b = 2^3 * 3 * 5^2 * a. The GCD of 3 * 5 * a and 2^3 * 3 * 5^2 * a, is 3 * 5 * a = 15a. Answer: B. Similar questions to practice: https://gmatclub.com/forum/which-of-the ... 67600.html (OG) https://gmatclub.com/forum/which-of-the ... 81173.html (GMAT Prep) https://gmatclub.com/forum/which-of-the ... 19455.html https://gmatclub.com/forum/which-of-the ... 67145.html https://gmatclub.com/forum/which-of-the ... 21900.html https://gmatclub.com/forum/what-is-the- ... 30418.html https://gmatclub.com/forum/if-x-and-y-a ... 74924.html https://gmatclub.com/forum/what-is-the- ... 42878.html Hope it helps.

egmat · Answer

When we say something is the GCD of two numbers, it must divide  BOTH  numbers.

Let's examine your suggestion for option B:
- You suggest: Let  a = 40 
- Then  4a = 160 
- And  15a = 600

Now here's the problem:  Does  160  divide  600 ?
 600 ÷ 160 = 3.75  (not a whole number!)

So  160  doesn't divide  600 , which means  4a  cannot be the GCD!

Why is option B different from D and E?

The  fundamental issue  is that  4  is not a factor of  15 . No matter what value of  a  you choose:
-  15a = 15 × a 
-  4a = 4 × a

For  4a  to divide  15a , we would need  4  to divide  15 , but  15 = 3 × 5  (no factor of  4 !).

In contrast, for options D and E: 
- Option D: We can make  40b  divide  15a  by choosing appropriate values
- Option E: We can make  15a  divide  40b  by choosing appropriate values

Key takeaway:  Option B is impossible because  4  doesn't divide  15 , so  4a  can never divide  15a , regardless of what  a  is.

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