How many two digit integers are there such that the product of their two digits is 24?
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08/11/2022
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Find the sum of all positive two-digit integers that are divisible by each of their digits. Solution 1Let our number be , . Then we have two conditions: and , or divides into and divides into . Thus or (note that if , then would not be a digit).If we ignore the case as we have been doing so far, then the sum is .Solution 2Using casework, we can list out all of these numbers: Solution 3To further expand on solution 2, it would be tedious to test all two-digit numbers. We can reduce the amount to look at by focusing on the tens digit. First, we cannot have any number that is a multiple of . We also note that any number with the same digits is a number that satisfies this problem. This gives We start from each of these numbers and constantly add the digit of the tens number of the respective number until we get a different tens digit. For example, we look at numbers and numbers . This heavily reduces the numbers we need to check, as we can deduce that any number with a tens digit of or greater that does not have two of the same digits is not a valid number for this problem. This will give us the numbers from solution 2.Solution 4In this solution, we will do casework on the ones digit. Before we start, let's make some variables. Let be the ones digit, and be the tens digit. Let equal our number. Our number can be expressed as . We can easily see that , since , and . Therefore, . Now, let's start with the casework.Case 1: Since , . From this, we get that satisfies the condition.Case 2: We either have , or . From this, we get that and satisfy the condition.Case 3: We have . From this, we get that satisfies the condition. Note that was not included because does not divide .Case 4: We either have or . From this, we get that and satisfy the condition. was not included for similar reasons as last time.Case 5: We either have or . From this, we get that and satisfy the condition.Continuing with this process up to , we get that could be . Summing, we get that the answer is . A clever way to sum would be to group the multiples of together to get , and then add the remaining .-bronzetruck2016 See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. How many twoThere are four 2-digit positive integers whose product of the two digits is 24 (38, 46, 64, and 83).
How many 2 digit integers are there?There are total 101 numbers from 0 to 100. To count all the 2 digit whole numbers, we have to remove all the 1 digit and 3 digit numbers from the count of 1st 100 numbers. Hence, there are total 90 two-digit whole numbers.
How many twoCounting the above digit we get a total of 17. So, there are 17 two-digit numbers whose sum of digits is a perfect square. Note: Perfect squares are those numbers which are formed when any number is multiplied by itself.
What is a 2 digit product?The simplest case is when two numbers are not too far apart and their difference is even, for example, let one be 24 and the other 28. Find their average: (24 + 28)/2 = 26 and half the difference (28 - 24)/2 = 2.
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