2020

Number Theory Level 2

Positive integers 0 ≤ a 1 < a 2 < a 3 < ⋯ < a n 0 \le a_1 < a_2 < a_3 < \cdots < a_n are such that

2 a 1 + 2 a 2 + 2 a 3 + ⋯ + 2 a n = 2020 2^{a_1}+2^{a_2}+2^{a_3}+\cdots+2^{a_n} = 2020

What is the value of n n ?


The answer is 7.

Transformation Tang by transformation tang , 17, China

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4 solutions

Luke Smith
Jul 27, 2020

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Nachiket Kanore
Mar 8, 2020

2020 is expressed as unique powers of 2. Hence it represents its binary representation and n will be equal to number of set bits in 2020.

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2020 = 4 × 505 = 4 ( 1 + 8 + 16 + 32 + 64 + 128 + 256 ) 2020=4\times 505=4(1+8+16+32+64+128+256) . So n = 7 n=7 .

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DEC:(2020) is equal to BIN(11111100100),Thus ,n=7

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