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!
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.