Certainly! Let's address each problem step-by-step:
### Question 5:
The Highest Common Factor (HCF) of two numbers is given as 12. We need to determine which of the provided options cannot be the Least Common Multiple (LCM) of these two numbers. The options are:
(a) 72
(b) 64
(c) 156
(d) 216
For two numbers to have an HCF of 12, it means that 12 is the largest number that divides both numbers without leaving a remainder. The LCM of two numbers is related to their HCF through the relationship:
[tex]\[ \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \][/tex]
Given [tex]\( \text{HCF} = 12 \)[/tex], the LCM must be a multiple of 12 to satisfy the relationship above. We now check each option to see if it is a multiple of 12.
- Option (a) 72: [tex]\(72 \div 12 = 6\)[/tex] (Since 6 is an integer, 72 is a multiple of 12)
- Option (b) 64: [tex]\(64 \div 12 = 5.3333\)[/tex] (Since 5.3333 is not an integer, 64 is not a multiple of 12)
- Option (c) 156: [tex]\(156 \div 12 = 13\)[/tex] (Since 13 is an integer, 156 is a multiple of 12)
- Option (d) 216: [tex]\(216 \div 12 = 18\)[/tex] (Since 18 is an integer, 216 is a multiple of 12)
The number which is not a multiple of 12 and therefore can never be the LCM of two numbers with an HCF of 12 is:
Option (b) 64
### Question 6:
Given three numbers [tex]\(P, Q,\)[/tex] and [tex]\(R\)[/tex] with the conditions:
1. The LCM of [tex]\(P\)[/tex] and [tex]\(Q\)[/tex] is [tex]\(Q\)[/tex].
2. The LCM of [tex]\(Q\)[/tex] and [tex]\(R\)[/tex] is [tex]\(R\)[/tex].
We need to find the LCM of [tex]\(P, Q,\)[/tex] and [tex]\(R\)[/tex]. Given the conditions:
1. [tex]\( \text{LCM}(P, Q) = Q \)[/tex]: This implies [tex]\(Q\)[/tex] is a multiple of [tex]\(P\)[/tex], since the LCM of two numbers is the smaller number when one is a multiple of the other.
2. [tex]\( \text{LCM}(Q, R) = R \)[/tex]: This implies [tex]\(R\)[/tex] is a multiple of [tex]\(Q\)[/tex], since the LCM of two numbers is the smaller number when one is a multiple of the other.
Thus, we can deduce:
[tex]\[ R \][/tex] is a multiple of [tex]\(Q\)[/tex], and [tex]\(Q\)[/tex] is a multiple of [tex]\(P\)[/tex]. When [tex]\(P\)[/tex], [tex]\(Q\)[/tex], and [tex]\(R\)[/tex] are such that one is a multiple of another down the line, the total LCM of [tex]\(P, Q,\)[/tex] and [tex]\(R\)[/tex] could either be [tex]\( P, Q, R\)[/tex], or some average if neither qualifies plainly due to other general constraints.
Given the conditions and processing through the logical placement:
[tex]\[ \boxed{\frac{P+Q+R}{3}} \][/tex]
This is the result of combining the conditions provided and considering the LCM function across the definitions provided for [tex]\(P, Q,\)[/tex] and [tex]\(R\)[/tex].
So, the correct answer for Question 6 is:
Option (d) [tex]\(\frac{P+Q+R}{3}\)[/tex]
### Summary:
The final answers are:
5. (b) 64
6. (d) [tex]\(\frac{P+Q+R}{3}\)[/tex]
