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Find the remainder when \(8\cdot10^{18}+1^{18}\) is divided by 9.

 May 25, 2024
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We can solve this problem using the divisibility rule for 9. This rule states that a number is divisible by 9 if the sum of its digits is divisible by 9.

 

Divisibility Rule for 9: A number (like 8*10^18 + 1^18) is divisible by 9 if the sum of its digits is divisible by 9.

 

Simplifying the Expression:

 

8*10^18 can be written as 800,000,000,000,000,000 (eight hundred quadrillion).

 

The sum of the digits in 800,000,000,000,000,000 is 8 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 8 = 16.

 

Remainder When Divided by 9:

 

We need to find the remainder when 8*10^18 + 1^18 (which has a digit sum of 16) is divided by 9.

 

Since 18 is a multiple of 2, 1^18 will always be 1. So, we can focus on the divisibility of 800,000,000,000,000,000 (with a digit sum of 16).

 

We can find the remainder by subtracting the nearest multiple of 9 that is less than 16: 16 - 9 (nearest multiple of 9 less than 16) = 7.

 

Therefore, the remainder when 8*10^18 + 1^18 is divided by 9 is 7.

 May 25, 2024

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