Find the H.C.F. of 108, 288 and 360.

### Concept & Strategy When dealing with medium-sized numbers, prime factorization remains a reliable method to find the Highest Common Factor (H.C.F.). By factoring each number into primes, the H.C.F. is simply the product of the lowest power of every common prime factor. ### Step-by-Step Solution **Step 1: Perform prime factorization.** * 108 = 4 $\times$ 27 = $2^2 \times 3^3$ * 288 = 32 $\times$ 9 = $2^5 \times 3^2$ * 360 = 8 $\times$ 9 $\times$ 5 = $2^3 \times 3^2 \times 5$ **Step 2: Identify common prime factors and their lowest powers.** * The prime number 2 is common to all. The powers are $2^2$, $2^5$, $2^3$. The lowest is $2^2$. * The prime number 3 is common to all. The powers are $3^3$, $3^2$, $3^2$. The lowest is $3^2$. * The prime number 5 is only in 360, so it is ignored. **Step 3: Multiply the lowest powers.** H.C.F. = $2^2 \times 3^2$ H.C.F. = 4 $\times$ 9 = 36. ### Exam Strategy & Shortcut Use the **Difference Method** combined with option elimination. The difference between 288 and 360 is 72. The H.C.F. must be a factor of 72. Check the smallest number (108). 108 is not divisible by 72. The next largest factor of 72 is 36 (since 72 / 2 = 36). Check 108: 108 / 36 = 3. Since 36 divides the smallest number perfectly, and is a factor of the difference between the larger two, 36 is our H.C.F. ### Common Pitfall A common error is to include the non-common factor 5 in the calculation or to mistakenly select the highest power of 2 ($2^5$), which would result in a number far too large to be a factor of 108. ### Final Answer **Therefore, the correct answer is 36.**