How Many Digits Are There? No Calculators Allowed.

Algebra Level 2

How many digits are there in the expansion of 100 1 2018 = ? \large1001^{2018}=?

6055 6035 6025 6045 6024

Hana Wehbi by Hana Wehbi , 51, USA

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2 solutions

Lucas Machado
Mar 5, 2018

3×1018+1 ,because we have (10^3+1)^2018

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But why is it 3 × 2018 + 1 3 \times 2018 + 1 ?

Hana Wehbi - 3 years, 3 months ago

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Because 10^1 has two digits.So 10^n have n+1 digits

LUCAS MACHADO - 3 years, 3 months ago

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Sorry by the english

LUCAS MACHADO - 3 years, 3 months ago

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@Lucas Machado – But we have 1 0 3 2018 10^{3^{2018}} ; so few details will explain the role of 3 3 . I see your point, it is correct but missing minor details. Thank you.

Hana Wehbi - 3 years, 3 months ago
Hana Wehbi
Mar 6, 2018

100 1 2 = 100 200 1 ⟹ 2 × 3 + 1 = 7 1001^2= 100 \ 200 \ 1\implies 2\times 3+1=7 digits.

100 1 3 = 100 300 300 1 ⟹ 3 × 3 + 1 = 10 1001^3= 100 \ 300 \ 300 \ 1\implies 3\times 3+1= 10 digits and so on...

Thus, 100 1 n has ( 3 × n + 1 ) digits ⟹ n = 2018 , 100 1 2018 1001^n \text { has } (3\times n + 1) \text { digits} \implies n= 2018, 1001^{2018} has 3 × 2018 + 1 = 6055 3\times 2018 +1=6055 digits.

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