Determine the HCF of the following numbers: \( \begin{array}{ll}\text { (1) } 36 \text { and } 54 & \text { (2) } \\ \text { (3) } 120 \text { and } 154 & \text { (4) } 96 \text {. } \\ \text { (5) } 126 \text { and } 294 \\ \text { (5) } 132 \text { and } 864 & \text { (6) } 300 \text { and } 360 \text {. } \\ \text { (7) } 480 \text { and } 432 & \text { (8) } \\ \text { (9) } 102,204 \text { and } 408 & \text { (10) } 144,216 \text { and } 525 \text {. } 312 .\end{array} \)
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To find the HCF (Highest Common Factor) of the given pairs, we can break them down as follows: 1. The HCF of 36 and 54 is 18. 2. The HCF of 120 and 154 is 2. 3. The HCF of 96 is 96 itself. 4. The HCF of 126 and 294 is 18. 5. The HCF of 132 and 864 is 12. 6. The HCF of 300 and 360 is 60. 7. The HCF of 480 and 432 is 48. 8. The HCF of 102,204 and 408 is 102. 9. The HCF of 144,216 and 525 is 9. Now, for the additional information: The concept of HCF has been used since ancient times, even by the ancient Greeks who studied geometry and number theory! They employed methods like Euclid's algorithm, which is still popular today for computing the HCF. So, next time you calculate it, remember you’re following in the footsteps of mathematical legends! In real-world applications, HCF is especially useful in simplifying fractions, dividing resources, or even optimizing production processes. For example, if you’re splitting food portions among friends, using the HCF helps ensure everyone gets an equal share without leftovers! It's a handy trick for anyone hosting a potluck!
