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An army contingent of 612 members is to march behind an army band of 48 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
The maximum number of columns in which both groups can march is 12.
To find the maximum number of columns in which both groups can march, we need to find the greatest common divisor (GCD) of the two numbers:
612 and 48.
The GCD of 612 and 48 can be found using the Euclidean algorithm or by listing the factors of each number.
Listing the factors of 612:
1, 2, 3, 4, 6, 9, 12, 17, 18, 19, 34, 36, 51, 68, 102, 153, 204, 306, and 612.
Listing the factors of 48:
1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
From the lists, we can see that the greatest common divisor (GCD) of 612 and 48 is 12.
So, the maximum number of columns in which both groups can march is 12.
Highest Common Factor and Least Common Multiple
The highest common factor (HCF) and least common multiple (LCM) are two important concepts in number theory.
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Highest Common Factor (HCF):
- The HCF of two or more numbers is the largest number that divides each of them without leaving a remainder.
- For example, the HCF of 12 and 18 is 6 because it is the largest number that divides both 12 and 18 evenly.
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Least Common Multiple (LCM):
- The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers.
- For example, the LCM of 4 and 6 is 12 because it is the smallest number that is divisible by both 4 and 6.
To find the HCF and LCM of a set of numbers, you can use various methods such as prime factorization, division method, or using the Euclidean algorithm.
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These methods provide systematic ways to calculate the HCF and LCM of numbers, which are useful in various mathematical problems and applications.
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