4. The L.C.M. and H.C.F. of two numbers are 360 and 10 respectively. If one of the numbers is 40, what is the other number?

5. The H.C.F. of 84 and 192 is 12. Find their L.C.M.

Question

High School

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Answer :

EmmaGraceJohnson EmmaGraceJohnson · Answer 1 · 2024-04-09T10:38:38-04:00

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.