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Updated on: 23 Jan 2019, 04:12
Originally posted by EgmatQuantExpert on 12 Dec 2018, 03:35.
Last edited by EgmatQuantExpert on 23 Jan 2019, 04:12, edited 3 times in total.
Last edited by EgmatQuantExpert on 23 Jan 2019, 04:12, edited 3 times in total.
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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35%
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Variations in Factorial Manipulation - Exercise Question #2
What is the highest power of 54 that can divide 46!?
To solve question 3: Question 3
To read the article: Variations in Factorial Manipulation
What is the highest power of 54 that can divide 46!?
- A. 0
B. 7
C. 12
D. 21
E. 42
To solve question 3: Question 3
To read the article: Variations in Factorial Manipulation
12 Dec 2018, 09:51
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EgmatQuantExpert
54= 3^3*2
Now lets find the number of 3's is [46/3]+[46/9]+[46/27] = 15+5+1 = 21
So, no. of 3^3 in 46! = 21/3 = 7
Also no. of 2's will be obviously greater than 3^3's ..
So, highest power of 54 that can divide 46! = 7
Answer B
14 Dec 2018, 21:51
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Solution
Given:
- • We are given a factorial value, 46! which is getting divided by a divisor 54.
To find:
- • The highest power of 54 that can divide 46!
Approach and Working:
As 54 is a composite number, first we need to prime factorize it in terms of the prime factors and their corresponding powers.
- • \(54 = 2^1 * 3^3\)
Next, we need to find the individual instances of 2 and 3.
- • Instances of 2: \(\frac{46}{2^1} + \frac{46}{2^2} + \frac{46}{2^3} + \frac{46}{2^4} + \frac{46}{2^5} = \frac{46}{2} + \frac{46}{4} + \frac{46}{8} + \frac{46}{16} + \frac{46}{32} = 23 + 11 + 5 + 2 + 1 = 42\)
• Similarly, instances of 3: \(\frac{46}{3^1} + \frac{46}{3^2} + \frac{46}{3^3} = \frac{46}{3} + \frac{46}{9} + \frac{46}{27} = 15 + 5 + 1 = 21\)
Now, the number 54 is formed by one instance of 2 and three instances of 3.
- • From 21 instances of 3’s, number of \(3^3\) can be formed \(= \frac{21}{3} = 7\)
• Hence, number of possible pairs of \(2^1\) and \(3^3\) = minimum (42, 7) = 7
Hence, the correct answer is option B.
Answer: B
