1. How many non-empty subsets containing only prime numbers does the set {1, 2, 3, 4, 5, 6, 7} have?

2. What two-digit number has exactly 9 factors?

3. How many factors of 1200 can be divided by 15 without a remainder?

4. When the positive integers with exactly three positive divisors are listed in ascending order, what is the fifth number listed?

5. N is the smallest positive integer that is divisible by every 1-digit positive integer. How many positive factors does N have?

6. Call a positive integer kinda-prime if it has a prime number of positive integer divisors. If there are 168 prime numbers less than 1000, how many kinda-prime positive integers are there less than 1000?

Guest Apr 12, 2019

edited by
Guest
Apr 12, 2019

#1**+2 **

1. How many non-empty subsets containing only prime numbers does the set {1, 2, 3, 4, 5, 6, 7} have?

We have four primes

The number of subsets formed by choosing any prime = C(4, 1) = 4

The number of subsets formed by choosing any two of the primes = C(4, 2) = 6

The number of subsets formed by choosing any 3 of the primes = C(4,3) = 4

And...if we are allowed an improper subet.....one more containing all the primes

So

4 + 6 + 4 + 1 = 15 subsets

Note that this is just the sum of the elements of the 4th row of Pascal's Triangle - 1 =

2^4 - 1 = 15

CPhill Apr 12, 2019

#2

#3**+2 **

3. How many factors of 1200 can be divided by 15 without a remainder?

1200 factors as

120 * 10 =

15 * 8 * 2 * 5 =

15 * 2^3 * 2 * 5 =

15 * 2^4 * 5

So....the number of factors of 2^4 * 5 = (4 + 1) (1 + 1) = 5 * 2 = 10

So

10 factors of 1200 are divisible by 15

CPhill Apr 12, 2019

#4**+2 **

4. When the positive integers with exactly three positive divisors are listed in ascending order, what is the fifth number listed?

The first number will be an even square....the others wll be odd

4 = 3 divisors

9 = 3 divisors

25 = 3 divisors

49 = 3 divisors

121 = 3 divisors

CPhill Apr 12, 2019

#5**+2 **

5. N is the smallest positive integer that is divisible by every 1-digit positive integer. How many positive factors does N have?

1 2 3 4 5 6 7 8 9

1 2 3 2^2 5 2*3 7 2^3 3^2

We pick off the unique integers along with their highest powers...so....

2^3 * 3^2 * 5^1 * 7^1 = 2520

The number of positive factors = ( 3 + 1) (2 + 1) (1 + 1) ( 1 + 1) = (4)(3)(2)(2) = 48 positive factors

CPhill Apr 12, 2019