Understanding Divisibility by 4
To determine if a number is divisible by 4, we only need to consider the last two digits of that number. If the last two digits form a number that is divisible by 4, then the entire number is divisible by 4.
Possible Digits
- The digits available are 2, 5, 6, and 7.
- Any combination of these digits can be used to form a four-digit number.
Finding Valid Last Two Digits
We will explore the combinations of the last two digits using the provided digits:
- Possible pairs: 25, 26, 27, 52, 56, 57, 62, 65, 67, 72, 75, 76
Now, we check which of these pairs are divisible by 4:
- 25 ÷ 4 = not divisible
- 26 ÷ 4 = not divisible
- 27 ÷ 4 = not divisible
- 52 ÷ 4 = divisible
- 56 ÷ 4 = divisible
- 57 ÷ 4 = not divisible
- 62 ÷ 4 = not divisible
- 65 ÷ 4 = not divisible
- 67 ÷ 4 = not divisible
- 72 ÷ 4 = divisible
- 75 ÷ 4 = not divisible
- 76 ÷ 4 = divisible
Valid pairs: 52, 56, 72, 76
Total Valid Combinations
- Each valid pair (last two digits) can be combined with the remaining digits (2 others) in the first two positions.
- Total combinations for each valid pair: 2! = 2 (since there are 2 remaining digits).
Calculating Total Outcomes
- Number of valid last two digits = 4
- Number of ways to arrange the first two digits = 2
Total valid four-digit numbers = 4 (valid last two digits) * 2 (arrangements) = 8
Calculating Total Possible Four-Digit Numbers
- Total four-digit combinations using digits 2, 5, 6, 7 = 4! = 24.
Final Probability
- Probability = Total valid outcomes / Total possible outcomes = 8 / 24 = 1/3.
Hence, the probability that a four-digit number formed from the digits 2, 5, 6, and 7 is divisible by 4 is 1/3.