Prove the following (how to tell if a number is divisible by 3)

This thread has 2 parts.

The first part consists of a problem I'm going to ask brains more intelligent than mine to help me solve.

The second part consists of my wanting to know whether a person who has a BA in math who is this helpless when it comes to proofs should bother with grad school.

The first part.

Prove that if

(x1 + x2 + ... + xn) % 3 = 0

where xi is an integer such that 0 <= xi <= 9

then the number whose digits equal those xi's, in that particular order, is also divisible by 3.

This relates to that shorthand way most of us learned in school to tell if a number is divisible by 3. For example, how do we tell if the number 23520 is divisible by 3? We add up its digits and if the sum is divisible by 3, then so is the number. In this case 2 + 3 + 5 + 2 + 0 = 12, which means that 23520 is also divisible by 3.

****

And now back to the second part of this post: given that I have a BA in math, and given that I spent almost 2 hours while I was driving on the highway trying to mentally solve this problem and couldn't do it, does it mean that I am not grad school material? Or am I trying to solve a problem that the average BA in math probably wouldn't be able to solve?

Re: Prove the following (how to tell if a number is divisible by 3)

Part 1)

Let N be a number represented (in decimal) by the digits

What is the numerical value of this number?

What is mod 3 ?

Part 2) It will be difficult to be successful as a grad. student in math without being able to do proofs. With that said: If you continue to work on proofs & get the right help, you can still get the hang of doing proofs.

Re: Prove the following (how to tell if a number is divisible by 3)

Quote:

Originally Posted by

**SammyS** Part 1)

Let N be a number represented (in decimal) by the digits

What is the numerical value of this number?

10000x1 + 1000x2 + blah blah blah

Quote:

What is

mod 3 ?

1

So we end up with a summation with xi's times 10^n-i (or whatever). And then we have to prove that that summation mod 3 is going to give us zero given that mod 3 of the sum of the xi's is zero. That's the part I can't do.

Re: Prove the following (how to tell if a number is divisible by 3)

Ok, we want to see if the number is divisible by 3.

Since is in the units column, its value is .

Since is in the 10s column, its value is .

Since is in the 100s column, its value is .

Continuing in this fashion we find that is in the s column, so its value is .

Also note that every power of 10 will have one remainder when divided by 3 (since they can be written as 9 + 1, 99 + 1, 999 + 1, etc.

So the value of is

Each of the terms is divisible by 3, so the number will only be divisible by 3 if the sum of the remaining terms is divisible by 3.

So the number is divisible by 3 if is divisible by 3.

Re: Prove the following (how to tell if a number is divisible by 3)

Quote:

Originally Posted by

**mathem** 10000x1 + 1000x2 + blah blah blah

1

So we end up with a summation with xi's times 10^n-i (or whatever). And then we have to prove that that summation mod 3 is going to give us zero given that mod 3 of the sum of the xi's is zero. That's the part I can't do.

So, N = 10000x_1 + 1000x_2 + blah blah blah.

And 10000x_1 + 1000x_2 + blah blah blah ≡ x_1 + x_2 + ... x_n (mod 3)

Therefore, N ≡ x_1 + x_2 + ... x_n (mod 3).

...

Re: Prove the following (how to tell if a number is divisible by 3)

Quote:

Originally Posted by

**Prove It** Ok, we want to see if the number

is divisible by 3.

Since

is in the units column, its value is

.

Since

is in the 10s column, its value is

.

Since

is in the 100s column, its value is

.

Continuing in this fashion we find that

is in the

s column, so its value is

.

Also note that every power of 10 will have one remainder when divided by 3 (since they can be written as 9 + 1, 99 + 1, 999 + 1, etc.

So the value of

is

Each of the

terms is divisible by 3, so the number will only be divisible by 3 if the sum of the remaining terms is divisible by 3.

So the number

is divisible by 3 if

is divisible by 3.

That makes sense. I just couldn't think of the (10^k - 1) trick.

Re: Prove the following (how to tell if a number is divisible by 3)

and just to quickly address the second part of my post,

The proof is clearly very simple but if you can't think of the trick used to get from A to B (ie: throw in -xi + xi) then you are pretty much ****ed.

This is the reason why I never bothered with grad school.

If these things don't jump at me, and become obvious only if I see them explained somewhere, then it probably means I would struggle in grad school not so much grasping the material per se but doing the out of the box thinking necessary to get the work done.

Just saying.