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? 

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.

1. 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.

2. 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.

• Prime Factorization Method:

  1. Find the prime factorization of each number.
  2. Multiply the common factors raised to the highest powers to find the HCF.
  3. Multiply all the prime factors, each raised to the highest power, to find the LCM.

• Division Method (also known as the Euclidean Algorithm):

  1. Take two numbers and find their remainder when divided.
  2. Replace the larger number with the remainder and repeat until the remainder is zero.
  3. The divisor at this point is the HCF of the numbers.
  4. To find the LCM, divide the product of the numbers by their HCF and multiply by the HCF.

These methods provide systematic ways to calculate the HCF and LCM of numbers, which are useful in various mathematical problems and applications.