# Find the sum of all the natural numbers not divisible by three between one and 3¹⁰⁰⁰⁰⁰

What is the easiest way to find the sum of all the numbers

$1, 2, 3, ..., 3¹⁰⁰⁰⁰⁰$ that are not divisible by 3,

possibly by using a handheld calculator and avoiding computer processing?

• Find the sum of all the numbers and then subtract the sum of the multiples of 3. – B. Goddard Mar 6 '17 at 22:54
• A reminder that $1+2+3+\dots+n = n(n+1)/2$ – JMoravitz Mar 6 '17 at 22:58
• Do you have a handheld calculate with roughly 66,000 digits? What would it even mean to calculate this without an electronic computer? – DanielV Mar 7 '17 at 0:03
• Hmm, closer to 95,000 digits. – DanielV Mar 7 '17 at 1:06

Hints: 1) How many numbers in that range are multiples of 3? 2) If you divide each of those numbers by 3, you get 1, 2, ..., k. What is the sum of those numbers? 3) Now add the numbers from 1 to $3^{100000}$ and subtract the sum above.
• The last number in the sequence that is divisible by 3 is clearly $3^{100000}$! – rogerl Mar 8 '17 at 2:06