Suppose the 435 members of the House of Representatives are placed on committees consisting of more than2 members but fewer than 30 members. Each committee is to have an equal number of members and each member is to be on only one committee.

a. What size committees are possible?
b. How many committees are there of each size?

The number of members in each committee must divide 435.

So let's us factorize 435 to get:
435=3*5*29
So the possible committee sizes are 3, 5 and 29.
Finally, divide 435 by the committee size to get the number of committees corresponding to each size.

a. To find the possible sizes of committees, we need to find factors of 435.

First, let's find the prime factorization of 435:
435 = 3 * 5 * 29

Next, let's list out all possible combinations of these prime factors:
- Factors of 435: 1, 3, 5, 15, 29, 87, 145, 435

Now, we need to find the factors within the given range of more than 2 members but fewer than 30 members. So, we can discard 1, 3, and 435 from the list.

The possible sizes of committees are: 5, 15, and 29.

b. To find the number of committees of each size, we need to divide the total number of members (435) by the size of each committee.

For a committee size of 5:
Number of committees = Total number of members / Size of committee
Number of committees = 435 / 5
Number of committees ≈ 87

For a committee size of 15:
Number of committees = Total number of members / Size of committee
Number of committees = 435 / 15
Number of committees ≈ 29

For a committee size of 29:
Number of committees = Total number of members / Size of committee
Number of committees = 435 / 29
Number of committees ≈ 15

So, there are approximately 87 committees of size 5, 29 committees of size 15, and 15 committees of size 29.