To find the relationship between the Highest Common Factor (H.C.F) and the Least Common Multiple (L.C.M) of the numbers 10 and 25, let's break down the solution step-by-step.
### Step 1: Identify the Numbers
We are given the numbers 10 and 25.
### Step 2: Find the Highest Common Factor (H.C.F)
The H.C.F. (or GCD - Greatest Common Divisor) of two numbers is the largest number that divides both of them without leaving a remainder.
For the numbers 10 and 25, the H.C.F is:
[tex]\[ \text{H.C.F}(10, 25) = 5 \][/tex]
### Step 3: Calculate the Least Common Multiple (L.C.M)
The L.C.M of two numbers is the smallest number that is a multiple of both.
There is a relationship between the L.C.M and H.C.F of two numbers given by the formula:
[tex]\[ \text{L.C.M}(a, b) \times \text{H.C.F}(a, b) = a \times b \][/tex]
Using this formula, we can find the L.C.M.:
[tex]\[ \text{L.C.M}(10, 25) = \frac{10 \times 25}{\text{H.C.F}(10, 25)} \][/tex]
[tex]\[ \text{L.C.M}(10, 25) = \frac{250}{5} = 50 \][/tex]
### Step 4: Confirm the Relationship
We can now check that the relationship holds correctly.
From the calculation above, we have:
[tex]\[ \text{H.C.F}(10, 25) = 5 \][/tex]
[tex]\[ \text{L.C.M}(10, 25) = 50 \][/tex]
Therefore, the relationship is:
[tex]\[ \text{L.C.M}(10, 25) \times \text{H.C.F}(10, 25) = 10 \times 25 \][/tex]
[tex]\[ 50 \times 5 = 250 \][/tex]
[tex]\[ 250 = 250 \][/tex]
Thus, the relationship between the H.C.F and L.C.M of the numbers 10 and 25 is validated.
### Final Answer
- H.C.F of 10 and 25 is 5.
- L.C.M of 10 and 25 is 50.
These values confirm the relationship [tex]\( \text{L.C.M}(10, 25) \times \text{H.C.F}(10, 25) = 10 \times 25 \)[/tex].
