# CAT Online Live Classes

Number Theory - Factors and Factorials

### Concepts discussed on July 8, 2017

#### Number Theory - Factors and Factorials

Welcome to the live session of Number Theory - Factors and Factorials. The concepts and questions discussed on July 8, 2017 during the live session follow the embedded video. The video link next to each concept will take you to the start of the part of the video where that specific concept is discussed. Click the video embedded below to watch the entire session.

#### Easy Questions

1. What is the smallest natural number that has exactly 12 factors?

2. What is the highest power of 8 that divides 100!?

3. What is the highest power of 12 that divides 45!?

4. What is the product of all factors of 144?

5. #### Medium Questions

6. What is the smallest natural number A such that 10A is a perfect square and 15A is a perfect cube?

7. In how many ways can we write 7560 as a product of two coprime numbers?

8. What is the largest number with exactly 4 factors that divides 28!?

9. How many factors of 24 * 35 * 56 * 73 are also multiples of 1800?

10. N is a two-digit number such that n +1 is not a factor of n!. How many different values can N take?

11. N2 has 45 factors. How many factors does N have?

12. How many values can N take if N! is a multiple of 1210 but not 1212?

13. Decimal representation of n! ends in exactly 14 zeroes. How many values can n take?

14. #### Tough Questions

15. In how many ways can we have two coprime natural numbers adding up to 1210?

16. Set S = {2, 3, 4, 5, ….100}. How many two-element subsets S exist such that the product of the elements in the subset has 5 or fewer factors?