The square root of the product of the positive integers a and b is 14,

Question

The square root of the product of the positive integers a and b is 14, which of the following could equal a + b?

I. 35
II. 53
III. 100

(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III

generis · Answer

I think that "a b" is supposed to be a+b.

Equation and factors 
The square root of the product of  positive  integers a and b is 14

\sqrt{ab}=14   - square both sides
  ab = 14^2 = 196

List the factors (a,b) of 196 
1 * 196
2 * 98
4 * 49
7 * 28
14 * 14

Options 
Using factor list (a, b) above, are there values of  a  and  b  that sum to the value in each option?

I. 35: a = 7, b = 28. (28+7) = 35. That works

II. 53: a = 4, b = 49. (4 + 49) = 53. That works

III. 100: a = 2, b = 98. (2 + 98) = 100. That works

Answer  E

Bunuel  , is that guess (about a typo in the prompt) correct?

Bunuel · Answer

Yes. Edited the typo. Thank you.

akadiyan · Answer

Square root of product of a and b is 14.

So product of a and b is 196.

Lets check the answer choices

A. 35 - 7*28 = 196 
B. 53 - 4*49 = 196
C. 100 - 2*98 = 196

All three options works.

Option E is the answer

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ScottTargetTestPrep · Answer

We are given that √(ab) = 14; thus, we see that ab = 14^2 = 196. Factoring 196 into product pairs, we have:

196 = 2 x 98 = 4 x 49 = 28 x 7

We see that the sum, a + b, could be 100 if a = 2 and b = 98; it could be 53 if a = 4 and b = 49; and, finally, it could be 35 if a = 28 and b = 7.

Answer: E

push12345 · Answer

ab=2*2*7*7

A+b= 35(28,7)
=53(49,4)
=100(98,2)
All three cases possible

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Nitintiwari · Answer

I think “B “ is the answer

√a * √b =14 
Because 14 is only divisible by 7 and 2 and 1 
For 53 -> √49 =7 -> a=49 , √4=2 -> b=4 ,a+b=53 and 7*2=14

Other. Case may not be possible

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MathRevolution · Answer

The square root of the product of the positive integers a and b is 14, therefore, a * b = 196

=> Possible combinations: 1 * 196 , 2 * 98, 4 * 49, 7 * 28, 14 * 14

a + b = 197, 100, 53, 35, 28

Answer E

bumpbot · Answer

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The square root of the product of the positive integers a and b is 14,

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