For all integers n such that n 1, the function f(n) is defined to be

Question

For all integers n such that n > 1, the function f(n) is defined to be the product of all the distinct prime factors of n. Which of the following values of n results in the smallest value of f(n)?

(A) 84
(B) 90
(C) 95
(D) 96
(E) 100

DrHuber · Answer

Don't get bogged down in the function language: the goal
is to compare each of the 5 numbers by finding the product of the prime factors
of each. That'll take some time, as it requires finding the distinct prime factors
of each number:
(A) 84 = 2²(3)(7), so 2 x 3 x 7 = 42
(B) 90 = 2(3²)(5), so 2 x 3 x 5 = 30
(C) 95 = 5(19), so 5 x 19 = 95
(D) 96 = 25(3), so 2 x 3 = 6
(E) 100 = 2²(5²), so 2 x 5 = 10
The smallest product is that of the prime factors of 96, choice (D).

M7e9s · Answer

Could you explain this answer please. 96=25(3)? I get 3x2x4x4, which would be 24.

Posted from my mobile device

Edit - I see the mistake now was writing 25 instead of 2^5. Answer makes sense!

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=v2yKYi-5as4

The function f(n) is defined to be the product of all the distinct prime factors of n

Lets take an example to understand the function

f(10) = 2*5 = 10 (As 2 and 5 are the distinct prime factors of 10)

Lets take each option choice and check which one gives the smallest value of f(n). 
 Tip: look only options which are multiples of higher powers of prime numbers

(A) 84 
84 = 21 * 4 =  2^2 * 3 * 7 
=> f(84) = 2 * 3 * 7 = 42

(B) 90 
 90 =  2 * 3^2 * 5 
=> f(90) = 2 * 3 * 5 = 30

(C) 95 
 95 = 5 * 19
=> f(95) = 5 * 19 = 95

(D) 96 
 96 = 32 * 3 =  2^5 * 3 
=> f(96) = 2 * 3 = 6 ( least till now )
 
 (E) 100  
 100 =  2^2 * 5^2 
=> f(100) = 2 * 5 = 10

So,  Answer will be D 
Hope it helps!

Watch the following video to MASTER Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

bumpbot · Answer

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For all integers n such that n > 1, the function f(n) is defined to be

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