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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: 9,355

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: 9,355

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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