The Remaining Number

Number Theory Level 2

Suppose you write down all the numbers from 1 to 1073741823. If you removed all the multiples of every odd prime number, starting from the largest prime less than 1073741823, what would be the last number on your new list?

536870912 There are no more numbers on the list 324782446 991026973 774840978 178946485

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2 solutions

After removing all multiples of all odd prime numbers, only powers of 2 2 remain. The maximum number of this form below 1073741523 1073741523 is 2 29 = 536870912 2^{29}=\boxed {536870912} .

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There is no way we are going to find the prime factorization of the answer choices.

We start looking for a pattern.

Removing all multiples of all odd prime numbers, we see only perfect powers of 2 remain. The value of the highest power of 2 that is below 1073741823 is 536870912.

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