To determine which of the given choices cannot be the Highest Common Factor (HCF) of two numbers whose Least Common Multiple (LCM) is 3600, we need to understand the relationship between LCM and HCF. For any two integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex], the product of their LCM and HCF is equal to the product of the numbers themselves:
[tex]\[ a \times b = \text{LCM}(a, b) \times \text{HCF}(a, b) \][/tex]
Given:
[tex]\[ \text{LCM}(a, b) = 3600 \][/tex]
We have a set of possible choices for HCF:
[tex]\[ \text{Choices} = [45, 15, 25, 50, 150] \][/tex]
We need to determine which of these choices cannot satisfy the condition.
1. If [tex]\(\text{HCF}(a, b) = 45\)[/tex]:
- Verify if there exist [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 3600 \times 45 = 162000 \][/tex]
- We must find integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 162000 \][/tex]
and
[tex]\[ \text{HCF}(a, b) = 45 \][/tex]
2. If [tex]\(\text{HCF}(a, b) = 15\)[/tex]:
- Verify if there exist [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 3600 \times 15 = 54000 \][/tex]
- We must find integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 54000 \][/tex]
and
[tex]\[ \text{HCF}(a, b) = 15 \][/tex]
3. If [tex]\(\text{HCF}(a, b) = 25\)[/tex]:
- Verify if there exist [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 3600 \times 25 = 90000 \][/tex]
- We must find integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 90000 \][/tex]
and
[tex]\[ \text{HCF}(a, b) = 25 \][/tex]
4. If [tex]\(\text{HCF}(a, b) = 50\)[/tex]:
- Verify if there exist [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 3600 \times 50 = 180000 \][/tex]
- We must find integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 180000 \][/tex]
and
[tex]\[ \text{HCF}(a, b) = 50 \][/tex]
5. If [tex]\(\text{HCF}(a, b) = 150\)[/tex]:
- Verify if there exist [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 3600 \times 150 = 540000 \][/tex]
- We must find integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[ a \times b = 540000 \][/tex]
and
[tex]\[ \text{HCF}(a, b) = 150 \][/tex]
After verifying all these choices, it turns out that none of these choices can be conclusively ruled out based on the available information.
Therefore, we can conclude that none of the given options cannot be the HCF.
The answer is:
```
None
```
