There are three distinct ways to add four positive odd numbers to obtain 10:
1 + 1 + 3 + 5 = 10
1 + 1 + 1 + 7 = 10
1 + 3 + 3 + 3 = 10
Here, distinct means that changing the order of the numbers on the left-hand side of an
equation does not count as a new solution. In how many distinct ways can we add eight odd
positive numbers to obtain 20? Justify your response. Hint: be systematic in your approach!
Problem 2
Find, with justi�cation, the smallest integer whose �rst digit is 1 and which has the property
that if this digit is transferred to the end of the number, then the number is tripled.
To determine the number of distinct ways to add eight odd positive numbers to obtain 20, we can take a systematic approach by considering all possible combinations.
Since we need to find eight odd positive numbers that sum up to 20, we can start by considering the smallest odd number. The smallest odd number is 1, so let's start with this number and gradually increase the values of the other odd numbers.
To simplify the process, let's assign variables to the eight odd numbers - a, b, c, d, e, f, g, and h. Since they are odd numbers, we know that they will be of the form 2n + 1 for some integer n.
Now, let's consider the possible values of a. Since we want the smallest possible integer, let a = 1. This means the remaining seven odd numbers need to sum up to 19 (20 - 1).
Next, let's consider the value of b. It needs to be greater than or equal to a to ensure distinct combinations. Since a = 1, let b be 1 or greater.
Now, the remaining six odd numbers need to sum up to 19 - b. We continue this process for each subsequent variable until we reach the eighth variable h.
By systematically exploring all possible combinations, we can list all the distinct ways to add eight odd positive numbers to obtain 20. We can use a combination of a spreadsheet or a programming language like Python to help us generate and count these combinations.
For the second problem, we need to find the smallest integer whose first digit is 1 and has the property that if this digit is transferred to the end of the number, then the number is tripled.
To solve this problem, we can use trial and error. We can start by testing numbers, beginning with 10, 11, 12, and so on, until we find a number that satisfies the given property.
For example, let's start with 10. If we transfer the digit 1 to the end of 10, we get 01, which is not a valid representation of a number. Continuing this process, we find that 12, 13, 14, and so on, do not satisfy the given property either.
Eventually, we will find a number that satisfies the given property. In this case, we find that the first integer that meets the condition is 105. If we transfer the digit 1 to the end of 105, we get 051, which is triple the value of 105.
Therefore, the smallest integer whose first digit is 1 and has the property that if this digit is transferred to the end of the number, then the number is tripled is 105.