Question

The integer 12 has 5 one digit factors, 20 has 4 one digit factors, and together 12 and 20 have 9 one digit factors(some of which are repeats). What is the total number of one digit factors for all of the integers from 1 to 100, inclusive?

Answer 1

I WILL GIVE MEDALS if anyone can figure it out

Answer 2

A good way to do this without going nuts, is to look at it the other way around and ask, for each of the numbers 1 through 9, how many of their multiples are less than or equal to 100?

For the number "1", we can see that there are 100 of these by finding all x such that:

1 < = (1)(x) <= 100 -> x = [100/1] where [ ] is a math symbol called "greatest integer".

For "2" this is 50 -> x = [100/2]

For "3" this is 33 -> x = [100/3] because we "round down" 33 and 1/3 to 33

So, you just do this for the rest of 4 to 9.

Answer 3

Every number from 1 to 100 is divisible by 1. So count 100.
50 of the first 100 natural numbers are divisible by 2. So add 50.
There are 33 numbers that are multiples of 4. Add 33.
There are 25 multiples of 4. Add 25.
There are 20 multiples of 5. Add 20.
There are 16 multiples of 6. Add 16.
There are 14 multiples of 14. Add 14
There are 12 multiples of 8. Add 12.
There are 9 multiples of 9. Add 9 more.
Answer is 279.

Answer 4

So, if you do this carefully, you will get

281

for your answer.

Answer 5

I did it very quickly. Thanks, @tcarroll010

Answer 6

Here is my corrected answer, thanks to @tcarroll010
Every number from 1 to 100 is divisible by 1. So count 100.
50 of the first 100 natural numbers are divisible by 2. So add 50.
There are 33 numbers that are multiples of 4. Add 33.
There are 25 multiples of 4. Add 25.
There are 20 multiples of 5. Add 20.
There are 16 multiples of 6. Add 16.
There are 14 multiples of 7. Add 14
There are 12 multiples of 8. Add 12.
There are 11 multiples of 9. Add 11 more.
Answer is 281.

Answer 7

All good now, @Charlievicky ?

Answer 8

Thanks:)

Answer 9

uw! Good luck to you in all of your studies! @Charlievicky

Answer 10

It's interesting how sometimes one has to look at a problem almost backwards to get it. It helps one think creatively and critically.

Answer 11

I think so too:)