Let P = 36000. Let Q equal the sum of all the factors of 36000, not

Question

Let P = 36000. Let Q equal the sum of all the factors of 36000, not including 36000 itself. Let R be the sum of all the prime numbers less than 36000. Rank the numbers P, Q, and R in numerical order from smallest to biggest.

(A) P, Q, R

(B) P, R, Q

(C) Q, P, R

(D) R, P, Q

(E) R, Q, P

(A) P, Q, R

(B) P, R, Q

(C) Q, P, R

(D) R, P, Q

(E) R, Q, P

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chetan2u · Answer

Hi..
Let's first look at Q and R..

36000= 2^5*3^2*5^3 ...
Number of factors is (5+1)(2+1)(3+1)=72-1.. -1 is for 36000 itself.
So Q would be sum of just 71 numbers, but R, sum of all prime numbers would be much higher since PRIME numbers less than 36000 would be way higher.. so Q<R..

Let's check P and Q..
Sum of factors would surely be more than the NUMBER itself..
P is 36000..
Just 3 factors -- 18000, 12000 and 9000 will cross the number P..
So P<Q..

Ans P<Q<R

A

GaurSaini · Answer

P = 36000
Q = Sum of factors of 36000, excluding 36000 
R = Sum of all the prime numbers till 36000

prime factorization of 36000 = 2^5 * 3^2 * 5^3.
(5+1) * (2+1) * (3+1) = 72 -1 = 71 factors

1. P and Q 
Let's take a few biggest factors of Q - 18000, 12000, 9000, 6000 - This sum is greater than 'P'. => Q>P

2. Q and R
Biggest number in Q is 18000, but there would definitely be a few prime numbers between 18000 and 36000. Also Q has only 71 numbers, but prime numbers in the range 1- 36000 would be way more than 71. Hence sum of all these numbers starting from 1 would be more than Q. => R > Q

R>Q>P

solvewithsourabh · Answer

Let P = 36000. Let Q equal the sum of all the factors of 36000, not including 36000 itself. Let R be the sum of all the prime numbers less than 36000. Rank the numbers P, Q, and R in numerical order from smallest to biggest.

(A) P, Q, R

(B) P, R, Q

(C) Q, P, R

(D) R, P, Q

(E) R, Q, P

(A) P, Q, R
(B) P, R, Q
(C) Q, P, R
(D) R, P, Q
(E) R, Q, P

Hi Bunuel,
This is one of the best tricky questions on the forum.
I am unable to approach this question mathematically. Even while applying logic (mostly my guess work), I am unable get to conclusive deductions regarding weather Q>R or R>Q. I would love to go through your answer for this question.

Bunuel · Answer

P = 36,000.

Q = the sum of all the factors of 36000, not including 36000 itself.
 Let's list some of the factors from largest to smallest:
18,000; 
12,000; 
9,000;
...
The sum of just these first few numbers is already greater than 36,000. Hence, we can confidently say P < Q. We can estimate that Q is approximately 100,000 (actually, it's 91,764).

R = the sum of all the prime numbers less than 36000.
 Prime numbers are relatively common in smaller ranges; for example, there are 25 prime numbers among the first 100 integers. Although they occur less frequently as the numbers get larger, even if we assume that there is at least one prime per thousand, we'd have primes like the following:
≈ 35,000;
≈ 34,000;
≈ 33,000;
≈ 32,000;
... 
The sum of these primes is clearly more than 100,000. Hence, Q < R. In fact, there are 3,824 prime numbers less than 36,000 (which is roughly 1 prime per 10 numbers), and their sum is 64,771,067. As you can see R (64,771,067) is much, much larger than Q (91,764).

Therefore, P< Q < R.

Answer: A.

pesi · Answer

the question said the sum of all the factors of P = 36,000. The number of positive factors of p will always be equal to the number of negative factors of p hence sum should be equal to 0

Ze3rebz · Answer

In a Question like this is it correct to use a smaller number to reach the answer by comparison ??

I used the number 36 and found the values of P, Q and R.

The value of P=36
Prime factors of 36 are 2^2 * 3^2 so Q will be 1+ 2 +3 +4 +9 +6 +12 +18
To find R - sum of all Prime numbers under 36 are 2+ 3+ 5+ 7+ 11+ 13+ 17+ 19 ...... (already the largest without adding the rest)

So A is the Answer P<Q<R

gchandana · Answer

I think it is risky, and we can't compare exactly.
Depending on what number you assume, you might get a different answer.

If one assumes, say, 15, then it would be Q, P, R.

So, it's better to go with the other solutions mentioned here.

Hope it helps.

Kinshook · Answer

Let P = 36000. Let Q equal the sum of all the factors of 36000, not including 36000 itself. Let R be the sum of all the prime numbers less than 36000.

Rank the numbers P, Q, and R in numerical order from smallest to biggest.

P = 36000 = 2^5*3^2*5^3 
 Q = (1+2+2^2+2^3+2^4+2^5)(1+3+3^2)(1+5+5^2+5^3) - 36000 = (2^6-1)/(2-1)*(3^3-1)/(3-1)*(5^4-1)/(5-1) - 36000 = 91764

Number of factors of P = 6*3*4 - 1 = 71

But R is the sum of all prime numbers less than 36000 is much higher number.

P < Q < R

IMO A

Let P = 36000. Let Q equal the sum of all the factors of 36000, not

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Let P = 36000. Let Q equal the sum of all the factors of 36000, not

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards