If a and b are both factors of 50, which of the following must be true

Question

If a and b are both factors of 50, which of the following must be true?

(A) ab≥ 2
(B) a + b≤ 50
(C) ab is a factor of 100.
(D) ab is a factor of 2500.
(E) ab is even.

eakabuah · Accepted Answer

We are informed that a and b are factors of 50. We are to determine which of the answer choices must be true.
factors of 50 are {1,2,5,10,25,50}.
In A, we are given that ab≥2. This is not always true for the situation a=b=1, where ab=1.
In B, a+b≤50. This also not always true for the situation a=25, and b=50, whereby a+b=75.
In C, ab is factor of 100. This not always true for a=10 and b=25, where ab=250.
In E, ab is even. This is not necessary true for a=1, and b=5, where ab=5, which is not even.
In D, ab is a factor of 2500. This always true since taking the highest possible values that a and be can take, a=b=50, and computing ab=2500. In this case 2500 is a factor of 2500. Hence D must be true.

Answer is therefore D.

unraveled · Answer

50 has total 6 factors – 1, 2, 5, 10, 25 and 50.

Since nothing is mentioned about relationship between ‘a’ and ‘b’ following conditions are possible:

a. a > b
b. a = b
c. a < b
where 1 ≤ a ≤ 50 and 1 ≤ b ≤ 50.

Let’s test by invalidating ‘must be true’ for all possible conditions –

Attachment:

File comment: 50 has total 6 factors –

50 has total 6 factors.JPG [ 75.65 KiB | Viewed 11657 times ]

* - Since 2500 has total 15 factors which include all the 6 factors of 50, taking possible extreme values of ‘a’ and ‘b’ is sufficient.

Answer (D).

EncounterGMAT · Answer

If a and b are both factors of 50, which of the following must be true?

Factors of 50: 1, 2, 5, 10, 25 and 50

(A) ab≥ 2 - If a=1 and b=2, ab≥ 2 but if a=1 and b=1, then a<2.....no

(B) a + b≤ 50 - if a=50 and b=25, then no

(C) ab is a factor of 100. - if a=25 and b=10, then no

(D) ab is a factor of 2500. - if a=50 and b=25...yes ALWAYS

(E) ab is even. - if a=1 and b=5, then no

IMO answer is  option D

exc4libur · Answer

number.f(50)=(5^2,2)=(3)(2)=6 ; and  factors(50)=(1,50,2,25,5,10) ;  a,b=any=(1,50,2,25,5,10)

(A) ab≥ 2:  a,b=(1,1):ab=1  false
(B) a + b≤ 50:  a,b=(50,50):ab=2500  false
(C) ab is a factor of 100:  a,b=(50,25):ab=1250  false
(E) ab is even:  a,b=(1,1):ab=1  false

Answer (D)

Mohammadmo · Answer

Factors of 50:2*5²
Factors of 2500:2²*5⁴
So, all factors of 50 are factors of 2500 too.
Option D

Posted from my mobile device

edo9 · Answer

Prime factorization results to: 50 = 10*5 = 5*2*5
Thus, factors are: 1, 2, 5, 10, 25, 50
What must be true is that ab >= 2 since smallest factors are 1 and 2.
So the correct answer should be A.

snoep · Answer

If a and b are both factors of 50, which of the following must be true?

factors of 50 = 1, 2, 5, 10, 25, 50
some options were very easy to eliminate, the one that took time was A and D ...however, at the end , figured out

(A) ab≥ 2
a = 1, b = 1 
1>=2 Not True

(B) a + b≤ 50
50+50 <= 50 -> Not true

(C) ab is a factor of 100
50*50 is not a factor of 100 -> Not true

(D) ab is a factor of 2500
50*50 is a factor of 2500 -> True
1*1 is a factor of 2500 -> True
2 * 25 is a factor of 2500 -> True

(E) ab is even
25 * 5 is not even -> Not true
1*5 is not even -> Not true

D is the answer!

CareerGeek · Answer

Factors of 50 are {1, 2, 5, 10, 25, 50}
Note: a & b can take same values!

(A) ab≥ 2 —> NO  
(B) a + b≤ 50 —> NO  
(C) ab is a factor of 100. —> NO  
(D) ab is a factor of 2500. —> YES  
(E) ab is even. —> NO

IMO Option D

Kinshook · Answer

a is a factor of 50 
b is a factor of 50
ab is a factor of 50*50 = 2500

IMO D

ScottTargetTestPrep · Answer

Since a is a factor of 50 and b is also a factor of 50, then ab is a factor of 50 x 50 = 2500.

To eliminate the other choices,consider the factors of 50: 1, 2, 5, 10, 25, and 50. 
Choice A is incorrect because if a = 1 and b = 1, then ab = 1, and A is not true.
Choice B is incorrect because if a = 50 and b = 10, then a + b = 60 and B is not true.
Choice C is incorrect because if a = 25 and b = 10, then ab = 250, which is not a factor of 100.
Choice D is true because every product of two factors of 50 is also a factor of 50 x 50 = 2500. 
Choice E is incorrect because if a = 1 and b = 5, then ab = 5, which is odd, so E is not true.

Answer: D

mysterymanrog · Answer

Here is my approach, took me 1 min total:

A) False, consider the case of A=B=1
B) False, consider the case of A=50, B=1
C) False, consider the case of A=B=50. 
D) Correct. Even for the largest value possible of AB (when A=B=50), this still holds. 
E) False. Consider the case of A=B=1

CharlyCH · Answer

Hello. 
I think I read that factors can be any number and here, it is not written that A and B are integer or non negative. Thus, can we think that D is the less wrong answer? 
I am confused because sometime we see very twisted question and sometime when we think about every single possibility, it is not included in the answer.

Thanks in advance

egmat · Answer

Hi CharlyCH,

I can see exactly where the worry comes from, and it's a good instinct to check whether the answer holds for  every  possible value. But here the fix is a definition, not a trap.

On the GMAT, "factor" always means a positive integer.  When a question says  a and b are factors of 50 , it is telling you that a and b each come only from the set every solver above listed:  {1, 2, 5, 10, 25, 50} . Negatives, fractions, and decimals are simply not "factors" in GMAT language. So you do  not  need to stretch a and b to any real number - the question has already restricted them for you.

That matters because it means D is not the "least wrong" answer - it is genuinely  must-be-true . Every a and b you can pick is a divisor of  50 , and  50  =  2 · 52 . So:

- a divides  2 · 52 
- b divides  2 · 52 
-  therefore  ab divides ( 2 · 52 )( 2 · 52 ) =  22 · 54  =  2500

The largest ab can ever be is  50  ×  50  =  2500 , which is itself a factor of  2500 , and every smaller product lands inside  2500 's factors too. There is no legal (a, b) pair that breaks it.

On your broader confusion:  the GMAT is precise with these words on purpose. "Factor," "divisor," "multiple," and "integer" all carry fixed, positive-integer meanings unless the problem explicitly says otherwise (for example, "x is a number" or "y can be negative"). So the rule of thumb is the reverse of what you feared: only widen your possibilities when the wording invites you to. Here it doesn't, so trust D fully.

Answer: D

If a and b are both factors of 50, which of the following must be true

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If a and b are both factors of 50, which of the following must be true

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