Find the H.C.F. of 513, 1134 and 1215.

### Concept & Strategy For large numbers, prime factorization can be extremely time-consuming and error-prone. Instead, use the **Long Division Method** (also known as the Euclidean algorithm). The logic is simple: divide the larger number by the smaller number. Then, make the remainder the new divisor and the previous divisor the new dividend. Continue until the remainder is 0. The last divisor is the H.C.F. ### Step-by-Step Solution When finding the H.C.F. of three numbers, find the H.C.F. of any two numbers first, and then find the H.C.F. of that result and the third number. **Step 1: Find the H.C.F. of the two largest numbers, 1134 and 1215.** * Divide 1215 by 1134. 1215 = (1134 $\times$ 1) + 81. The remainder is 81. * Now divide the previous divisor (1134) by the remainder (81). 1134 = (81 $\times$ 14) + 0. The remainder is 0. The H.C.F. of 1134 and 1215 is 81. **Step 2: Find the H.C.F. of the result (81) and the remaining number (513).** * Divide 513 by 81. 513 = (81 $\times$ 6) + 27. The remainder is 27. * Now divide the previous divisor (81) by the remainder (27). 81 = (27 $\times$ 3) + 0. The remainder is 0. The final divisor is 27. ### Exam Strategy & Shortcut Use the **Difference Method** first to narrow down options. Difference between 1215 and 1134 is 81. The H.C.F. of all three numbers MUST be a factor of 81. Looking at the options, 27, 81, and 9 are all factors of 81 (eliminate 54). Test 81 against the remaining number 513: 513 / 81 = 6.33 (Not perfectly divisible). Test the next largest factor, 27: 513 / 27 = 19. Since 27 divides 513 perfectly, it is the H.C.F. This takes seconds compared to full long division. ### Common Pitfall A major pitfall is stopping after Step 1. Students often find the H.C.F. of the first two numbers (81) and immediately select it as the final answer, completely forgetting to test it against the third number (513). ### Final Answer **Therefore, the correct answer is 27.**