Even numbers are formed by talking at least two at a time from the numbers 0,…

To find the Highest Common Factor (H.C.F.) of all even numbers formed by taking at least two digits at a time from the set {0, 4, 8, 9}, we first identify the numbers that can be formed and then determine their common factors.

Step-by-Step Analysis
1. Form the even numbers:
To form an even number, the last digit must be 0, 4, or 8.

• Two-digit even numbers: 40, 80, 90, 48, 84, 94, 98 (Note: 0 cannot be the first digit).

• Three-digit even numbers: 480, 940, 490, 890, 498, etc.

2. Find the common factors:

• All these numbers are even, so they must be divisible by 2. This means 2 is a common factor.

• Now, check if a larger number can be the H.C.F. by examining a few examples:

• Take the numbers 98 and 40.

• The factors of 98 are 1, 2, 7, 14, 49, 98.

• The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40.

• The only common factors between 98 and 40 are 1 and 2.

Since the greatest common factor between even just two of these numbers (98 and 40) is 2, the H.C.F. of the entire set of possible even numbers must be 2.