Question

A magic number is a number that'll divide evenly every number which ends in itself, for instance, 2 is magic because 2 divides into every number finishing in 2(2,12,22,102 etc). Find a rule to determine whether a number is magical.

Answer 1

I'll give medals for every attempt.

Answer 2

if you want a guess i would say 2, 5, or any power of 5 or 10?

Answer 3

lets start with easy ones.
2 works
5 works
10 works

Answer 4

You're getting hot satellite73.

Answer 5

25 works

Answer 6

Yes, now, try to factor what you found. See if patterns emerge.

Answer 7

so i looks like powers of 5 or powers of 2*5

Answer 8

Yes, getting better and better.

Answer 9

2^n * 5*n

Answer 10

Almost there james. The powers don't have to be necessarily the same.

Answer 11

2^n * 5^n i meant :)

Answer 12

well not exactly. 2^n5^n=10^n so not all

Answer 13

2^p * 5^q

Answer 14

maybe 2^n5^m

Answer 15

There you are. :)

Answer 16

that looks good to me

Answer 17

Now some sort of proof, doesn't have to be rigorous...

Answer 18

satellite did all the work for me...i just tried to capitalize on his work

Answer 19

^^

Answer 20

guess my guess was a good guess, even if i guess it was just a guess

Answer 21

must prove that 2^n * 5^n mod n = 0

Answer 22

Lol.

Answer 23

exactly like saying that if you divide by a such a number you get a terminating decimal

Answer 24

james: generalize the numbers, as in, finishing in 5 for instance works because if a number is abc...10 you can make it abc...00 + 10 which factors in 5 ( abc.... + 1).

Answer 25

finishing in 10 for instance I meant.

Answer 26

abc...10 = a * 10^p + b * 10^(p-1) + ... + 10

so if you divide that by 10, you get:

a * 10^(p-1) + b * 10^(p-2) + ... + 1

Answer 27

Yes.

Answer 28

which is still an integer

Answer 29

Right. Now guys, expand the generalization to 2^n * 5^m.

Answer 30

(2^n * 5^m) = (abc...10) * (def...5)

on the right track?

Answer 31

wait...it's (abc...2) * (def...5) ?

Answer 32

Yep. Kind of. Done. Do you want another challenge? :)

Answer 33

(abc...2) * (def...5) will be a huge ugly series, but the constant term
will be 10, which is divisible by both 5 and 2

Answer 34

i am not very good at number theory! but that was an interesting problem :)

Answer 35

:) Glad you liked it. Shame satellite went to other stuff... Well, ok, here's a challenge, something not for now, but to think over and then come back ok? May I? :)

Answer 36

sure

Answer 37

If you tell a person to choose a number smaller than 60 (but positive), and then ask them to 1) divide it by 3 and tell you the remainder, 2) divide the remainder by 4 and tell you the remainder of that, 3) divide that remainder by 5 and tell the remainder of that.
Then, the number choosen is 40 * the first remainder + 45 * the second remainder + 36 * the third remainder mod 60.

Answer 38

a perfect number is any number whose factors ...add up to itself right?

Answer 39

I don't know? :(

Answer 40

yes, that's a perfect number. ana described magic numbers.

Answer 41

perfect numbers = 6, 28 etc.

Answer 42

Ana, are we supposed to prove that?

Answer 43

Yes.

Answer 44

If you want to see more about it, see: "Chinese Remainder Theorem".

Answer 45

you must be a math professor. you are smart :)

Answer 46

No. I'm just a student. But I'll take the compliment. hehe. You're smart too.

Answer 47

Enter the chat.