Final answer:
There are 5 positive integers that consist solely of the digits 1 and 3 and have a sum of digits equal to 12; these are 111111111111, 3111111111, 33111111, 333111, and 3333.
Step-by-step explanation:
Finding Positive Integers with Specific Conditions
To find the number of positive integers that satisfy the conditions that each digit is a 1 or a 3 and the sum of the digits is 12, we can look at combinations of 1s and 3s that add up to 12. Considering the smallest possible integer using the digits 1 and 3 that sums to 12 is 111111111111, and the largest is 3333, we need to find how many different combinations of these digits sum to 12. We can systematically list these combinations starting with the integer that has the most 3s (since 3 is the higher value digit) and progressively replacing a 3 with two 1s until we can no longer do so.
The combinations that satisfy the sum condition are:
-
- 111111111111
-
- 3111111111
-
- 33111111
-
- 333111
-
- 3333
Therefore, there are 5 positive integers that meet both conditions.