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The product of the largest two factors of a positive integer n is 16875. What is the difference between the largest positive integer that divides √n and the smallest odd factor greater than 1 of √n?
- a)2
- b)12
- c)13
- d)222
- e)223
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The product of the largest two factors of a positive integer n is 1687...
Given:
- n is an integer > 0
- Let a and b respectively be the two largest factors of n
- a * b = 16875
To Find: Difference between the largest positive factor that divides √n and the smallest odd factor of √n greater than 1.
- Largest positive factor that divides √n will be √n itself
- Smallest odd factor of √n greater than 1) = (smallest odd prime factor of √n)
Approach:
- To find the factors of √n, we first need to find the value of √n. For finding the value of √n, we need to find the value of n.
- We are given information about the product of the largest two factors of n
c. For finding the smallest prime factor of n, we need to find the smallest prime factor of 16875.
- This is because the prime factors of 16875 will only be the prime factors of n, as 16875 is the product of n with one of the factors of n.
Working out:
c. As the smallest prime factor of 16875 is 3, the second largest factor of n = n/3 =b
2. So, √n = 15
3. Largest positive factor that divides √n = √n = 15
4. Smallest odd factor of √n greater than 1 = smallest odd prime factor of √n = 3
5. So, Largest positive factor that divides √n - Smallest odd factor of √n greater than 1 = 15 – 3 = 12
3. Largest positive factor that divides √n = √n = 15
4. Smallest odd factor of √n greater than 1 = smallest odd prime factor of √n = 3
5. So, Largest positive factor that divides √n - Smallest odd factor of √n greater than 1 = 15 – 3 = 12
Answer B
Most Upvoted Answer
The product of the largest two factors of a positive integer n is 1687...
Understanding the problem:
Given that the product of the largest two factors of a positive integer n is 16875, we need to find the difference between the largest positive integer that divides n and the smallest odd factor greater than 1 of n.
Solution:
Finding the prime factorization of 16875:
16875 = 3 * 5^3 * 11
Finding the factors of n:
Since the largest two factors of n multiply to 16875, they must be 225 and 75. This means n = 225.
Finding the largest positive integer that divides n:
The largest positive integer that divides 225 is 225 itself.
Finding the smallest odd factor greater than 1 of n:
The smallest odd factor greater than 1 of n is 3.
Calculating the difference:
The difference between the largest positive integer that divides n and the smallest odd factor greater than 1 of n is:
225 - 3 = 222
Therefore, the correct answer is option B (222).
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