Final answer:
There are 9 positive integers less than one billion that are divisible by 9 and have all equal digits. These are the sequences from 111,111,111 to 999,999,999 with each digit ranging from 1 to 9.
Step-by-step explanation:
The question regards finding how many positive integers less than one billion are divisible by 9 and have all digits equal. A number with all equal digits is divisible by 9 if the sum of its digits is also divisible by 9. We can consider numbers with equal digits like 111,222,...,999,000,000, where each sequence of a digit is repeated until we reach just below one billion (999,999,999). To find the count, we'll determine which of these conform to our criteria. For example, 111,111,111 (sum of digits = 9) meets the criteria, while 222,222,222 (sum of digits = 18) also meets the criteria, and so on. We note that each number where digits sum to a multiple of 9 will meet the criteria.
Starting with 1 and ending with 9, the positive integer sequences that are permissible (since they're less than a billion and have a sum of digits divisible by 9) are: 111,111,111; 222,222,222; 333,333,333; 444,444,444; 555,555,555; 666,666,666; 777,777,777; 888,888,888; and 999,999,999. Therefore, there are 9 such numbers.