Number Theory questions -- please help!?

Question

Please show me an easy way to do these problems. And remember that I have no knowledge of number theory.

1. Determine the smallest positive integer n such that 5^7 divides n!

2. How many integers between 1000 and 10,000 are divisible by 7?

3. How many integers less than 1000 are divisible by 3 but not by 4?

4. How many integers between 0 and 10,000 are divisible by 3, 5 and 7?

Thanks!

Answer 1

1) so we know that 5^7 contains 7 multiples of 5 and that means that n! must also contain 7 multiples of 5 to make it divisble so the smallest integer must at least be the 7th multiple of 5 so each multiple divides by the 5; the 7th multiple of 5 is 35

so n= 35 n! = 35!

2) find the smallest integer divisble by 7 that is greater than 1000

1000/ 7 = 142.8 so the 143th mutliple of 7

now find greatest integer less than 10000

10000/7 = 1428.5 = 1428

so the number of integers is 1428- 143 + 1 (add one since you have to count end number as well; always to this for all counting questions involving counting number of numbers between initial and end number since you have to include them and normal subtraction only includes one of them )  = 1286 integers

3) so find all the ipositive ntegers that are divisble by 3 and 4 and subtract it from all those that are divisble by 3

divisble by 3 less than 1000  1000/ 3 = 333 + 1 = 334 integers (including zero)

now divisible by 3 and 4 means multiple of 12 so

1000/ 12 = 83 + 1 (including zero) = 84 integers by 12

so numbers by 3 and not 4 = 334 - 84 + 1 = 251 integers

4) since 3,5,7 are prime it means that a number that is divisble by all of them must have them as prime factors; smallest possible number is 3*5*7 = 105

and between 0-10000 10000/105 = 95 + 1 (including zero) =96 integers

hope this helps

Answer 2

1. ANSWER: n = 35

5, 10, 15, 25, 30, 35

5 has one factor of 5

10 has one factor of 5

15 has one factor of 5

20 has one factor of 5

25 has two factors of 5

30 has one factor of 5

35 has one factor of 5

In other words, 40! has 7 factors of 5.

2. 1,000 < 7n < 10,000

Find the inequality for which n lies in.

3. Infinity? Natural numbers perhaps?

Same concept as the second problem. However, this time, you subtract the multiples of 12.

4. Find the number of of multiples of 105 between 0 and 10,000.