Answer :
To solve these questions, we need to understand the concepts of L.C.M. (Least Common Multiple) and H.C.F. (Highest Common Factor).
Finding the Other Number Given L.C.M. and H.C.F.:
Given:
- L.C.M. = 360
- H.C.F. = 10
- One number is 40
We can use the relationship between two numbers [tex]a[/tex] and [tex]b[/tex] alongside their L.C.M. and H.C.F.:
[tex]\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b[/tex]
Let's denote the unknown number as [tex]x[/tex]. Thus the equation becomes:
[tex]360 \times 10 = 40 \times x[/tex]
Solving for [tex]x[/tex], we divide both sides by 40:
[tex]x = \frac{360 \times 10}{40}[/tex]
Simplifying the right side:
[tex]x = \frac{3600}{40} = 90[/tex]
Therefore, the other number is 90.
Finding the L.C.M. Given the H.C.F.:
Given:
- H.C.F. of 84 and 192 is 12
We use the same relationship again:
[tex]\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b[/tex]
Plug in the values we know:
[tex]\text{LCM}(84, 192) \times 12 = 84 \times 192[/tex]
Solving for LCM, divide both sides by 12:
[tex]\text{LCM}(84, 192) = \frac{84 \times 192}{12}[/tex]
Simplifying the right side:
[tex]\text{LCM}(84, 192) = \frac{16128}{12} = 1344[/tex]
Therefore, the L.C.M. of 84 and 192 is 1344.
