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# Nine digits => two numbers => largest product

## Question

Here are your digits: 1 2 3 4 5 6 7 8 9.

Using each of them exactly once,  form two numbers: for example, 314879 and 652.

What's the largest product two such numbers can have?

You don't need a computer

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in general the largest number produced by multiplication is achieved when the two numbers are close together.

so we want a 4 digit and a 5 digit number.

a b c d e

* f g h i

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the largest first digit occurs when a and f are 9 and 8. since we want them to be close, its better for a to be 8 and f to be 9.

the second largest digit will be 2 + 9*b +8*g. the largest value occurs when b is 7 and g is 6.

this puts the max value at 8300000 aprox. the next largest digit is 7*6 +9*c +8*h. the max value occurs when h is 4 and c is 5.

you can continue. the final answer is: 9642 *87531.

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we make:

98765432 as first number
and 1 as second number
and we will get the largest product
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we make:

98765432 as first number

and 1 as second number

and we will get the largest product

98765431 x 2 = 197530862 is bigger.

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9 x 87654321

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9 x 87654321

91 x 87654321 is bigger
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maybe 911 x 87654321 then?

Anyway, back to work.

Edited by kukupai
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9652 x 87431

(and running out of ideas )
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Sooooo close. Swap a single digit from each number and you have it.

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9642 x 87531 ???

Edited by kukupai
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Toying with a script to spit out the answers quickly, it seems this is the general pattern:

If the digits are dm..dn (contiguous) the answer looks like:

d n dn-3dn-5.. . * dn-1dn-2dn-4...

regardless of m and n. This works regardless of whether there is an even or odd number of digits.

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Toying with a script to spit out the answers quickly, it seems this is the general pattern:

If the digits are dm..dn (contiguous) the answer looks like:

d n dn-3dn-5.. . * dn-1dn-2dn-4...

regardless of m and n. This works regardless of whether there is an even or odd number of digits.

Nice!

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