How to Factor 1 Trillion into Two Numbers Without Zeroes?

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Discussion Overview

The discussion centers around the challenge of factoring 1 trillion into two numbers without any digits being zero. Participants explore various approaches, including integer and non-integer factorizations, while considering different interpretations of the term "trillion." The scope includes mathematical reasoning and exploratory problem-solving.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to factor 1 trillion into two numbers without zeroes.
  • Another participant suggests two factorizations, 262,144 x 3,814,697.266 and 5^12 x 2^12, but expresses uncertainty about their correctness, particularly noting that the second option contains zeroes.
  • A participant proposes that the riddle might refer to the British definition of "trillion" (10^18), offering factors of 2^18 and 5^18.
  • One participant asserts that 5^12 x 2^12 is the only factorization that avoids ending with zero, arguing that any factor containing both 2 and 5 would be divisible by ten.
  • Another participant clarifies that the requirement is for the numbers to have no zeroes at all, not just to avoid ending in zero.
  • A suggestion is made that the solution may involve non-integer factors or non-decimal systems.
  • Several participants calculate factors, noting that 2^12 and 5^12 both contain zeroes, which complicates their use.
  • One participant provides a factorization of 1 x 1717914916, later correcting it to 1 x 34771513724319.
  • Another participant mentions that if decimals are allowed, 262,144 x 3,814,697.265625 could work.
  • There is a call for clarification on whether integers are required, with one participant stating that factoring into integers would be impossible under the given constraints.

Areas of Agreement / Disagreement

Participants express differing views on whether the problem allows for decimal factors or requires integer factors. There is no consensus on the correct approach or solution, and multiple competing interpretations of the problem remain.

Contextual Notes

Some participants note that the definitions of "trillion" may vary (American vs. British), and there is ambiguity regarding the requirement for factors to be integers or decimals. The discussion reflects uncertainty about the constraints of the problem.

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

Thanks.
 
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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.
 
Cesius said:
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).
 
I'm American, and it's an American school so I'm pretty sure it mean 10^12.

Thanks though.
 
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.)
 
It's not that it can't end in zero, it just can't have any zeros in it.
 
Then some trick must be involved. Either you're allowed to use non-integer factors, or you're allowed to use non-decimal systems.
 
[itex]2^12= 4096[/itex] and [itex]5^12= 244140625[/itex], 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.
 
1*1717914916

Edit: on the second reading I see hamster already hinted at this solution.
 
  • #10
Borek said:
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"? ;)
 
  • #11
No idea what have happened, obviously some finger slip on my side. 1*34771513724319 looks much better.
 
  • #12
Cesius said:
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.
 
  • #13
We are allowed decimals. Thanks for the answers Astronuc! And thanks for the help everyone!
 
  • #14
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.
 
  • #15
Try this: Factor 611939 into two different integer factors. ;)

^^
 
  • #16
Cesius said:
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.
 

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