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**Roger Austin****Member**- Registered: 2022-01-02
- Posts: 2

Look at number 36. If you multiply the two individual digits and then the product by 2 you get 36.

So 36 = (3x6)x2

Is there any other number that can duplicate this? Namely ABCD--n digits = (AxBxCxDx--n digits) x2? If not why not? I found other numbers with different multipliers like:

144 = (1x4x4)x9 and 224 = (2x2x4)x14

Are these unique numbers?

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,514

hi Roger Austin

Welcome to the forum.

I think that there may be many such. This is how I'm going to start looking.

You seem to be happy to multiply by 9 or 14 etc so I'll assume all you want is [number] = [any multiple of] [product of digits]

So algebraically:

Find a b and n so that 10a + b = nab for whole numbers a, b and n. {this is the two digit version}

I'll see if I can find some more .

On my spreadsheet allowing a and n to be from {2,3,4,5,6,7,8,9} I only got one more: b = 4 and 24 = 2 x 4 x 3

Three digits may take longer.

Actually not much longer as it was quick to add 100 to my 10a + b.

I found four in the one hundreds:

a = 7, b = 5 , n = 5 175 = 1 x 7 x 5 x 5

a = 2, b = 8, n = 8 128 = 1 x 2 x 8 x 8

a = 3, b = 5, n = 9 135 = 1 x 3 x 5 x 9

a = 4, b = 4, n = 9 144 = 1 x 4 x 4 x 9

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Roger Austin****Member**- Registered: 2022-01-02
- Posts: 2

Thanks Bob. But is the multiplier 2 for 36 unique? I cannot find any other number.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,514

Unless I've completely misunderstood, it has to be unique. The number is 36 so the product begins 3 x 6 x <something> and has to result in 36. There's only one solution to this equation:

3 x 6 times [letter x] = 36 => 18x = 36 => x = 36/18 = 2

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**pamshaw****Member**- Registered: 2021-12-07
- Posts: 21

The problem here is the discussion is about numbers and multiplication. There is an example of the number 36 that if you multiply two numbers and then multiply the result with the two, you will get the number 36. The question that comes from here is there any number that follows the same pattern in any number of digits? If not, then why?

The discussion on this point is accepted with open arms in the community; the community answers the question with proof of the existence of certain other numbers on a variety of numerics.

a = 7, b = 5 , n = 5 175 = 1 x 7 x 5 x 5

a = 2, b = 8, n = 8 128 = 1 x 2 x 8 x 8

a = 3, b = 5, n = 9 135 = 1 x 3 x 5 x 9

a = 4, b = 4, n = 9 144 = 1 x 4 x 4 x 9

Starting with the acceptance that several digits can exist in the pattern discussed in the problem proves an abundant variable available that you can find. Though there was some difficulty in understanding the equation, the community did find the solution keeping the context that the number can be in any range and giving the result lies the numbers in the statement.

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