A positive integer n is completely divisible by 12 and 8. If √nlies between 5 and 8, exclusive, how many values of n are possible?a)0b)1c)2d)9e)10Correct answer is option 'B'. Can you explain this answer?

Question

A positive integer n is completely divisible by 12 and 8. If &radic;nlies between 5 and 8, exclusive, how many values of n are possible?a)0b)1c)2d)9e)10Correct answer is option 'B'. Can you explain this answer?

Answer

Understanding the Problem
We need to find positive integers n that are completely divisible by both 12 and 8, and whose square root lies between 5 and 8, exclusive.
Step 1: Finding the Least Common Multiple
- Since n must be divisible by both 12 and 8, we calculate the least common multiple (LCM).
- The LCM of 12 and 8 is 24. Thus, n must be a multiple of 24.
Step 2: Establishing the Range for n
- The condition √n lies between 5 and 8 means:
- 5 < √n='' />< />
- Squaring gives us:
- 25 < n='' />< />
Step 3: Finding Multiples of 24
- We now find the multiples of 24 within the range (25, 64).
- The multiples of 24 are: 24, 48, 72, etc.
- Within our range, we check:
- The first multiple of 24 greater than 25 is 48.
- The next multiple, 72, exceeds 64.
Step 4: Valid Values of n
- The only multiple of 24 that satisfies the condition is:
- n = 48
Conclusion
- Therefore, the only value of n that is completely divisible by 12 and 8 and falls within the specified range is 48.
- Hence, there is only 1 possible value of n.
Final Answer
The correct answer is option 'B', which means there is 1 value of n possible.

Answer

Understanding the Problem
We need to find a positive integer n that is completely divisible by both 12 and 8, and where the square root of n falls between 5 and 8, exclusive.
Step 1: Finding the Least Common Multiple (LCM)
- The LCM of 12 and 8 is calculated as follows:
- Prime factorization:
- 12 = 2^2 * 3
- 8 = 2^3
- LCM = 2^3 * 3 = 24
- Therefore, n must be a multiple of 24.
Step 2: Setting Up Inequalities
- We need to determine n such that 5 < √n='' />< />
- Squaring the inequalities gives:
- 25 < n='' />< />
Step 3: Finding Multiples of 24 within the Range
- Now, we need to find multiples of 24 that fall between 25 and 64.
- The multiples of 24 are: 24, 48, 72, etc.
- Focusing on the range:
- The first multiple of 24 greater than 25 is 48.
- The next multiple, 72, is outside our range since 72 > 64.
Step 4: Conclusion
- The only multiple of 24 that satisfies 25 < n='' />< 64='' is='' />
- Thus, there is only 1 value of n that meets all conditions.
Final Answer
The correct answer is option 'B', indicating there is only 1 possible value for n.

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