Math Is Fun Forum

Haiying Liao

Guest

patterns in multiplication table~ explain why it work this ?

Refer to the base 10 multiplication table. ..

the products of the diagonals of any 2 by 2 matrix are equal. Explain why.

Look at the 9s column in the multiplication table. The sum of the digits is always 9 . can you explain why ?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: patterns in multiplication table~ explain why it work this ?

Hi;

Because those numbers are all multiples of 9, therefore divisible by 9. All numbers that are divisible by nine have the sum of their digits divisible by nine:

Here is a proof.

http://www.pseudorandom.co.uk/2002/maths/divby9/

the products of the diagonals of any 2 by 2 matrix are equal. Explain why.

Algebraic notation proves this easily.

Let us refer to the 10 x 10 matrix in the usual way of r = row and c = column. We can say that any element or entry of the table is c*r.

Then any 2 x 2 will be of the form.

Now just multiply the two diagonals out.

Rearrange to see it easier.

That is all there is to it.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Haiying Liao

Guest

Re: patterns in multiplication table~ explain why it work this ?

Hi;

Because those numbers are all multiples of 9, therefore divisible by 9. All numbers that are divisible by nine have the sum of their digits divisible by nine:

Here is a proof.

http://www.pseudorandom.co.uk/2002/maths/divby9/

Can you explain more about this question? I dont really get it

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: patterns in multiplication table~ explain why it work this ?

Hi Haiying Liao;

Did you look at that proof? What part of the question is bothering you?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

hy

Guest

Re: patterns in multiplication table~ explain why it work this ?

Hi Haiying Liao;

Did you look at that proof? What part of the question is bothering you?

( or can you proof it by using the multiplication table ....like as we go down the column, the ones digit decrease by 1 each time and the tens digit increases by 1 each time......can you explain why ???  )

i dont really understand this part ...:o

It's possible to prove that this will always work - one possible proof is:

Let m be an n-digit number, and its digits in reverse order be d0, d1, up to dn-1, so each di is a whole number between 0 and 9 inclusive.

This implies that

m = d0 + 10d1 + 100d2 + 1000d3 + ... + 10n-1dn-1
⇔ m = (d0 + d1 + d2 + d3 + ... + dn-1) + (9d1 + 99d2 + 999d3 + ... + (a number with n-1 digits all of which are 9) dn-1)
⇔ m = (the sum of all m's digits) + 9(d1 + 11d2 + 111d3 + ... + (a number with n digits all of which are 1) dn-1)
so obviously, if m is divisible by 9, so is the sum of m's digits, and vice versa. Also, since 9 is divisible by 3, the same occurs for 3.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: patterns in multiplication table~ explain why it work this ?

Hi Haiying Liao;

One thing at a time. Are you okay with the proof on that page?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.