Find the H.C.F. and L.C.M. of $0.63$, $1.05$ and $2.1$.

### Concept & Formula To find the H.C.F. and L.C.M. of decimals, you must first convert them to equivalent numbers with the **same number of decimal places**. Once aligned, temporarily remove the decimal point, find the H.C.F. and L.C.M. of the resulting integers, and then place the decimal point back in the result at the same position. ### Step-by-Step Solution **Step 1: Align the decimal places.** * Given numbers: $0.63$, $1.05$, $2.1$ * The maximum number of decimal places is two (in $0.63$ and $1.05$). * Rewrite $2.1$ as $2.10$ so all numbers have two decimal places. * The aligned numbers are: $0.63$, $1.05$, $2.10$. **Step 2: Treat as integers.** * Remove the decimals to get: $63$, $105$, $210$. **Step 3: Calculate the H.C.F. of the integers.** * $63 = 3^2 \times 7$ * $105 = 3 \times 5 \times 7$ * $210 = 2 \times 3 \times 5 \times 7$ * The common prime factors are $3$ and $7$. * H.C.F.($63, 105, 210$) = $3 \times 7 = 21$. **Step 4: Calculate the L.C.M. of the integers.** * L.C.M. requires the highest powers of all prime factors ($2, 3, 5, 7$). * L.C.M.($63, 105, 210$) = $2^1 \times 3^2 \times 5^1 \times 7^1$ * L.C.M. = $2 \times 9 \times 5 \times 7 = 630$. **Step 5: Restore the decimal point.** * Since we originally aligned the numbers to two decimal places, insert the decimal point two places from the right. * Final H.C.F. = $0.21$ * Final L.C.M. = $6.30$ (or simply $6.3$) ### Exam Strategy & Shortcut A much faster method for alignment is to multiply all given numbers by $100$ to clear the decimals completely: $63$, $105$, and $210$. Find the H.C.F. ($21$) and L.C.M. ($630$) of these whole numbers. Then, simply reverse the initial operation by dividing your answers by $100$. $21 / 100 = 0.21$. $630 / 100 = 6.3$. ### Common Pitfall The most dangerous pitfall is failing to align the decimal places first. If you extract $63$, $105$, and $21$ directly from $0.63$, $1.05$, and $2.1$, your H.C.F. calculation for the integers will be $21$, which leads to $0.21$, but your L.C.M. will be $315$, leading to an incorrect final L.C.M. of $3.15$. Always pad with trailing zeros first. ### Final Answer **Therefore, the correct answer is H.C.F. = $0.21$, L.C.M. = $6.3$.**