Final answer:
To determine the probability that 12d4 is divisible by 12, we must use divisibility rules for 3 and 4. Only the digits 2 and 8 for d result in the number 12d4 being divisible by 12, leading to a probability of 1/5.
Step-by-step explanation:
To find the probability that the four digit integer 12d4 is divisible by 12, we need to consider the divisibility rules for 12. A number is divisible by 12 if it is divisible by both 3 and 4.
For divisibility by 3, the sum of the digits must be divisible by 3, and for divisibility by 4, the last two digits must be divisible by 4.
Since 4 is already given as the last digit, we only need to consider which digits (when placed in the tens place) make the last two digits divisible by 4.
The choices for d that satisfy this condition are 0, 2, 4, 6, and 8. Now, taking all the digits into account and applying the divisibility rule for 3, the sum of the digits 1 + 2 + d + 4 must also be divisible by 3.
The sums for each valid d option are:
- For d=0, sum is 7 which is not divisible by 3
- For d=2, sum is 9 which is divisible by 3
- For d=4, sum is 11 which is not divisible by 3
- For d=6, sum is 13 which is not divisible by 3
- For d=8, sum is 15 which is divisible by 3
Out of the digits 0-9, only 2 and 8 make the number 12d4 divisible by 12. Therefore, with 2 out of 10 possible digits working, the probability is
2/10
or
1/5.