A warm up that might be useful later:

Show that all integers greater than 17 can be written in the form 10a + 3b, where a and b are non-negative integers.

The main problem:

Noa is building triangular cup towers where the top row has 1 cup, and each other row has 1 more cup than the row above it. She defines an n-tower as a triangular tower using n cups, where n > 1.

For example, a 6-tower has 1 cup in the top row, 2 cups in the next row, and 3 cups in the bottom row. Note that there is no such thing as a 1-tower, which is just a single cup.

Noa has N cups. She builds a number of n-towers, using all N cups. If N = 12, she can build:

• 4 3-towers
• 2 6-towers
• 1 6-tower and 2 3-towers

In total, she can build n-towers for 2 different values of n. Note that she can't build a 10-tower, because there's no way to use the remaining 2 cups.

(a) Verify that when N = 18, Noa can build n-towers for 3 different values of n.

(b) Noa wants to build n-towers for 13 different values of n. What is the smallest possible value of N?