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12 Feb 2021, 05:30
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
33% (03:04) correct
67%
(03:03)
wrong
based on 61
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What is the number of ways in which the number 300300 can be written as the product of 2 positive factors, if these factor do not share any common factor but 1?
A. 8
B. 16
C. 24
D. 32
E. 64
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
A. 8
B. 16
C. 24
D. 32
E. 64
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
12 Feb 2021, 06:09
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Bunuel
The first step would be to find different factors.
300300 should have 300 as a factor => 300*1001
1001 should be a multiple of 11 as per the property of multiples of : sum of digits at even places-sum of digits at odd places should be divisible by 11.
1+0-(0+1)=0.
300*1001=300*11*91=300*11*7*13
\(300300=2^2*3*5^2*7*11*13\)
So we have 6 distinct prime factors and one more positive integer factor as 1.
Different ways to select two factors, say a*b, will be when power of a prime factor is completely in one of the a or b.
1) All 6 together : 1 way
2) 5 and 1 : ways to choose 5 out of 6 =6C5=6
3) 4 and 2 : ways to choose 4 out of 6 =6C4=15
4) 3 and 3 : ways to choose 3 out of 6 =6C3=20. But here the moment we choose one set of three, the other is already chosen. For example A*B and B* A will be same as both A and B have three factors. So the total = 20/2=10
Final answer =1+6+15+10=32.
OR
Let the two numbers be A and B
So A*B=300300
Now we have 6 prime factors, and they can be in any of the variables. So possibilities = 2*2*2*2*2*2 ( 6 times because there are 6 distinct prime factors)=64
But A*B=B*A, so half the numbers = 64/2=32
D
22 Nov 2022, 17:10
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Bunuel
Let's prime factorize 300300: 2^2 * 3 * 5^2 * 7 * 11 * 13
If the two factors are not to share any common factors, then one of the two factors must contain 2^2 (because if each factors contains one 2, then the two factors will have the common factor of 2). Similarly, one of the two factors must contain 5^2. So, the answer to the question will be the same if we consider the number 2 * 3 * 5 * 7 * 11 * 13.
Case 1. One of the factors is prime
When we choose the prime factor, the other factor will be the product of the remaining five numbers. We can choose the prime factor in 6 ways, so there are 6 ways to express the number as a product where one of the factors is prime.
Case 2. One of the factors is a product of two primes
If one of the factors is a product of two primes, then the other factor will be the product of the remaining four numbers. We can choose two primes from among six numbers in 6C2 = 6!/(2! * 4!) = (6 * 5)/2! = 15 ways.
Case 3. Both factors are a product of three primes
We can choose three primes from among six numbers in 6C3 = 6!/(3! * 3!) = (6 * 5 * 4)/3! = 20 ways. However, this is actually twice the number of ways to express the given number as a product where both factors are products of three primes, because choosing the three primes in the first factor and choosing the three primes in the second factor actually correspond to the same outcome (for instance, choosing 2^2, 3, and 5^2 corresponds to the product 300 * 1001, and so does choosing 7, 11, and 13). Thus, we can express the given number as a product where each factor is a product of three primes in 20/2 = 10 ways.
Case 4. One of the factors is 1
There is only one way to express the given number as a product where one of the factors is 1.
In total, we can express 300300 as a product of two relatively prime numbers in 6 + 15 + 10 + 1 = 32 ways.
Answer: D
