How to Factor 1 Trillion into Two Numbers Without Zeroes?

[Cesius]

Can anyone tell me how to factor 1 trillion in 2 numbers, and none of them have zeroes in them?

Thanks.

 

[Cesius]

The two numbers I got were 262,144 x 3,814,697.266 and 5^12 x 2^12. I'm not sure which one is right because the first one I'm not sure if it's exact and the second one technically has zeroes.

 

[rphenry]

The two numbers I got were 262,144 x 3,814,697.266 and 5^12 x 2^12. I'm not sure which one is right because the first one I'm not sure if it's exact and the second one technically has zeroes.

Usually in riddles of this nature the "two numbers" are assumed to be integers.

Perhaps the riddle meant the British definition of "trillion", 10^18, in which case your factors would be 2^18 and 5^18, (262144 and 3814697265625).

 

[Cesius]

I'm American, and it's an American school so I'm pretty sure it mean 10^12.

Thanks though.

 

[hamster143]

It's straightforward to prove that 5^12 x 2^12 is the only factorization into two factors such that neither of the two ends with zero. (Because any factor that has a 2 and a 5 would be divisible by ten.)

 

[Cesius]

It's not that it can't end in zero, it just can't have any zeros in it.

 

[hamster143]

Then some trick must be involved. Either you're allowed to use non-integer factors, or you're allowed to use non-decimal systems.

 

[HallsofIvy]

$2^12= 4096$ and $5^12= 244140625$, both of which have 0s so [math](2^12)(5^12([/math] and with any other two factors at least one would have to have factors of "2" and "5" and so be a multiple of 10.

 

[Borek]

1*17179149₁₆

Edit: on the second reading I see hamster already hinted at this solution.

 

[sjb-2812]

1*17179149₁₆

Isn't this sum wrong anyway, as 1 trillion (either long or short scale), 10 and 16 are all co-prime to 2, so it cannot end in an odd "digit"? ;)

 

[Borek]

No idea what have happened, obviously some finger slip on my side. 1*3477151372431₉ looks much better.

 

[Astronuc]

The two numbers I got were 262,144 x 3,814,697.266 and 5^12 x 2^12. I'm not sure which one is right because the first one I'm not sure if it's exact and the second one technically has zeroes.

262,144 x 3814697.265625 works if one is allowed decimals.

 

[Cesius]

We are allowed decimals. Thanks for the answers Astronuc! And thanks for the help everyone!

 

[HallsofIvy]

It would have helped if you had told us that to begin with! You said "factor one trillion" and that usually means into integer factors.

 

[FeDeX_LaTeX]

Try this: Factor 611939 into two different integer factors. ;)

^^

 

[Astronuc]

We are allowed decimals. Thanks for the answers Astronuc! And thanks for the help everyone!

You're welcome. I was wondering if one had to have integers which would be impossible given the constraints. I just start dividing 5¹² by 2 until I found a number without zeros.

On the other hand, one can also divide 2¹² by 5's, in which case only one division is needed 2¹²/5 = 819.2 and 48828125 = 5¹¹.