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.

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 –

* - 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).
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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

\(number.f(50)=(5^2,2)=(3)(2)=6\); 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)

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

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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.

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!

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

(A) ab≥ 2 —> NO [(a, b) = (1, 1)]
(B) a + b≤ 50 —> NO [(a,b) = (25, 50), (50, 50) etc.,]
(C) ab is a factor of 100. —> NO [(a, b) = (50, 50) etc.]
(D) ab is a factor of 2500. —> YES [Maximum value of ab = 50*50 = 2500, which is still a factor of 2500]
(E) ab is even. —> NO [(a, b) = (1, 1), (5, 25) etc.]

IMO Option D

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

IMO D
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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
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