Answer :
Let's solve the problems one at a time, explaining each step clearly.
The product of two numbers is 2080 and their H.C.F. is 16. Find their L.C.M.
The relationship between two numbers [tex]a[/tex] and [tex]b[/tex], their Highest Common Factor (H.C.F.), and Least Common Multiple (L.C.M.) is given by:
[tex]a \times b = \text{H.C.F.} \times \text{L.C.M.}[/tex]
Here, the product of the numbers is 2080, and the H.C.F. is 16. By substituting into the formula:
[tex]2080 = 16 \times \text{L.C.M.}[/tex]
Dividing both sides by 16 gives:
[tex]\text{L.C.M.} = \frac{2080}{16} = 130[/tex]
So, the L.C.M. of the two numbers is 130.
The product of two numbers is 2880 and their L.C.M. is 480. Find their H.C.F.
Again, using the relationship:
[tex]a \times b = \text{H.C.F.} \times \text{L.C.M.}[/tex]
Substitute the given values:
[tex]2880 = \text{H.C.F.} \times 480[/tex]
Dividing both sides by 480 gives:
[tex]\text{H.C.F.} = \frac{2880}{480} = 6[/tex]
So, the H.C.F. of the two numbers is 6.
The H.C.F. of two numbers is 29 and their L.C.M. is 435. If one of the numbers is 145, find the other.
Using the same relationship:
[tex]145 \times b = 29 \times 435[/tex]
Calculate the right side first:
[tex]29 \times 435 = 12615[/tex]
Therefore, the equation becomes:
[tex]145 \times b = 12615[/tex]
Solving for [tex]b[/tex]:
[tex]b = \frac{12615}{145} = 87[/tex]
So, the other number is 87.
Find the H.C.F. and L.C.M. of the following numbers:
(i) 30 and 54
H.C.F.: To find the H.C.F., determine the prime factors:
- 30 = 2 \times 3 \times 5
- 54 = 2 \times 3^3
- Common factors are [tex]2[/tex] and [tex]3[/tex].
- H.C.F. = 2 \times 3 = 6
L.C.M.: The L.C.M. is found by taking the highest power of each prime that appears in the factorizations:
- L.C.M. = 2^1 \times 3^3 \times 5 = 270
(ii) 174 and 261
H.C.F.: Find prime factors:
- 174 = 2 \times 3 \times 29
- 261 = 3 \times 29 \times 3
- Common factors are [tex]3[/tex] and [tex]29[/tex].
- H.C.F. = 3 \times 29 = 87
L.C.M.:
- L.C.M. = 2^1 \times 3^2 \times 29 = 522
(iii) 90 and 230
H.C.F.:
- 90 = 2 \times 3^2 \times 5
- 230 = 2 \times 5 \times 23
- Common factors are [tex]2[/tex] and [tex]5[/tex].
- H.C.F. = 2 \times 5 = 10
L.C.M.:
- L.C.M. = 2^1 \times 3^2 \times 5^1 \times 23 = 2070
(iv) 36 and 48
H.C.F.:
- 36 = 2^2 \times 3^2
- 48 = 2^4 \times 3^1
- Common factors: 2^2, 3
- H.C.F. = 2^2 \times 3 = 12
L.C.M.:
- L.C.M. = 2^4 \times 3^2 = 144
Therefore, we've calculated the H.C.F. and L.C.M. for each set of numbers.
