Question

How many positive integers less than or equal to 2000 have an odd number of factors?

Answer 1

Hint: look at the factorizations of the first 10 positive whole numbers and see which number has an odd number of factors

Answer 2

I'll look and see... wait a second

Answer 3

ok

Answer 4

From 1-10 would only be the number 1; srry I kinda had to go do something for a sec

Answer 5

you're fine

Answer 6

you sure only 1 has an odd number of factors?

Answer 7

and 4

Answer 8

what else

Answer 9

and 9

Answer 10

what do you notice

Answer 11

Can't really think of anything

Answer 12

Can you give me a hint?

Answer 13

1, 4, 9, ...

hmm

Answer 14

what kind of sequence is that

Answer 15

if you're not sure, look at the next ten numbers (11 through 20) and take note which numbers have an odd number of factors

Answer 16

Sounds like +3, +5, maybe then +7

Answer 17

Yup this works:)

Answer 18

that's one way to look at it, but there's another

Answer 19

the extended sequence is 1, 4, 9, 16, 25, 36, 49, ...

Answer 20

each number is a perfect _____

Answer 21

Oh I didn't notice that

Answer 22

so all you have to do is count the number of perfect squares less than 2000

use this list: http://www.mathwarehouse.com/arithmetic/numbers/list-of-perfect-squares.php

or

you can take the square root of 2000 to get 44.7213595499958

this means that 44^2 = 1936 is the largest perfect square that is less than 2000

anything higher (like 45^2) is going to be over 2000

Answer 23

On this problem I'm supposably not supposed to use a list, so if 44^2 is the largest perfect square does that mean that I should count all the perfect squares 44^2 and below?

Answer 24

yep, so 1^2, 2^2, 3^2, ..., 41^2, 42^2, 43^2, 44^2 all work

Answer 25

Would 1^3 work too though?

Answer 26

1^2 = 1 is a perfect square

Answer 27

1 only has 1 factor (itself), so it has an odd number of factors

Answer 28

1^3 is still 1, but technically not the same

Answer 29

because if you cubed 2, you would get 2^3 = 8, but 8 doesn't have an odd number of factors

Answer 30

Oh, so 44 positive integers or is it more complicated than what I'm thinking?

Answer 31

yep 44 is your answer and maybe you are, but that's ok, you're thinking about the problem

Answer 32

Thanks:)

Answer 33

yw