A positive multiple of 11 is good if it does not contain any
even digits in its decimal representation.
(a) Find the number of good integers less than 1000.
(b) Determine the largest such good integer.
(c) Fix b ≥ 2 an even integer. Find the number of positive integers less
than b
3 which are divisible by b+ 1 and do not contain any even digits
in their base b representation. (This is the natural generalization of
part (a) with 10 replaced by b.)
11,33,55,77,99,121,319,517,539,...,979
(a) To find the number of good integers less than 1000, we need to check each number from 1 to 1000 to see if it meets the criteria of being a positive multiple of 11 and not containing any even digits in its decimal representation.
First, we need to check for positive multiples of 11. We can start by finding the smallest positive multiple of 11 which is greater than or equal to 1000 using division. Divide 1000 by 11 and round up to the nearest integer to get this value: 91. So, we will check numbers in the range 11 to 91 (inclusive) to see if they are multiples of 11.
Next, we need to check if these numbers do not contain any even digits in their decimal representation. To do this, we can look at each digit of the number and check if it is even or odd. If any digit is even, we exclude that number.
Count the numbers from 11 to 91 that have only odd digits.
(b) To find the largest good integer, we need to search for the largest multiple of 11 that does not contain any even digits in its decimal representation.
Starting from 1000, count down by subtracting 11 until we find a number that meets the criteria. Check if each number has only odd digits until we find the largest number.
(c) To find the number of positive integers less than b^3 that are divisible by b+1 and do not contain any even digits in their base b representation, we need to follow a similar process as part (a) but with base b instead of base 10.
First, find the smallest positive integer greater than or equal to b^3 that is divisible by (b+1). Divide b^3 by (b+1) and round up to the nearest integer. This value will be our upper limit for checking.
Check all numbers in the range from 1 to that upper limit. For each number, convert it to base b and check if it contains any even digits. Exclude any number that has even digits.
Count the numbers that pass these criteria.