2, 4, 8, 16, 32, etc
When will the next number in this sequence be a power of 10. (10, 100, 1000, 10,000, 100,000, etc)
If you can reach this number, what significance does it have?
This is impossible. No power of two can ever even be divisible by ten, since no power of two is divisible by five.
Oh and one thing you may wish to observe: the last digit of powers of two will continue to follow the pattern: 2, 4, 8, 6, 2, 4, 8, 6…
This is impossible. No power of two can ever even be divisible by ten, since no power of two is divisible by five.
Oh and one thing you may wish to observe: the last digit of powers of two will continue to follow the pattern: 2, 4, 8, 6, 2, 4, 8, 6…
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Never. Every power of ten can be divided by 5, but no power of two is divisible by five.
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If you were allowed to run the sequence backward, you would find one match: 2^0 = 10^0, or 1 = 1.
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We can formalize the situation in the following way:
We want to find a number m such that
2^m = 10^n, for some number n.
But this is precisely
2^m = 10^n = (2 * 5)^n = 2^n * 5^n. Dividing the 2^n from both sides gives:
2^(m-n) = 5^n.
This is impossible by the fundamental theorem of arithmetic; in other words, we cannot represent the same number as a two different products each containing different primes.
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