# Factor the repunit ## R_{6}=111111 ## into a product of primes

• Math100
In summary: Hence ## 111,111 = 3 \cdot 7 \cdot 11 \cdot 13 \cdot 37 ##.In summary, the repunit ## R_{6} = 111111 ## can be factored into a product of primes as ## 3 \cdot 7 \cdot 11 \cdot 13 \cdot 37 ##. This can be shown using the divisibility rules for 3, 7, 11, and 13.
Math100
Homework Statement
Factor the repunit ## R_{6}=111111 ## into a product of primes.
Relevant Equations
None.
Consider the repunit ## R_{6}=111111 ##.
Then ## R_{6}=111111=1\cdot 10^{5}+1\cdot 10^{4}+1\cdot 10^{3}+1\cdot 10^{2}+1\cdot 10^{1}+1\cdot 10^{0} ##.
Note that a positive integer ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ## where ## 0\leq a_{k}\leq 9 ## is
divisible by ## 7, 11 ##, and ## 13 ## if and only if ## 7, 11 ##, and ## 13 ## divide the
integer ## M=(100a_{2}+10a_{1}+a_{0})-(100a_{5}+10a_{4}+a_{3})+(100a_{8}+10a_{7}+a_{6})-\dotsb ##.
This means ## a_{0}=a_{1}=a_{2}=a_{3}=a_{4}=a_{5}=1 ##.
Thus ## M=(100+10+1)-(100+10+1)=0 ##.
Since ## 7, 11 ##, and ## 13 ## divide ## 0 ##, it follows that ## 7, 11 ##, and ## 13 ## divide the repunit ## R_{6} ##.
Observe that the sum of digits in ## R_{6} ## is ## 1+1+1+1+1+1=6 ##.
This means ## 3\mid R_{6} ##.
Thus ## R_{6}=111111=3\cdot 7\cdot 11\cdot 13\cdot 37 ##.
Therefore, a product of primes in ## R_{6} ## is ## 3\cdot 7\cdot 11\cdot 13\cdot 37 ##.

Delta2
Math100 said:
Homework Statement:: Factor the repunit ## R_{6}=111111 ## into a product of primes.
Relevant Equations:: None.

Consider the repunit ## R_{6}=111111 ##.
Then ## R_{6}=111111=1\cdot 10^{5}+1\cdot 10^{4}+1\cdot 10^{3}+1\cdot 10^{2}+1\cdot 10^{1}+1\cdot 10^{0} ##.
Note that a positive integer ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ## where ## 0\leq a_{k}\leq 9 ## is
divisible by ## 7, 11 ##, and ## 13 ## if and only if ## 7, 11 ##, and ## 13 ## divide the
integer ## M=(100a_{2}+10a_{1}+a_{0})-(100a_{5}+10a_{4}+a_{3})+(100a_{8}+10a_{7}+a_{6})-\dotsb ##.
This means ## a_{0}=a_{1}=a_{2}=a_{3}=a_{4}=a_{5}=1 ##.
Thus ## M=(100+10+1)-(100+10+1)=0 ##.
Since ## 7, 11 ##, and ## 13 ## divide ## 0 ##, it follows that ## 7, 11 ##, and ## 13 ## divide the repunit ## R_{6} ##.
Observe that the sum of digits in ## R_{6} ## is ## 1+1+1+1+1+1=6 ##.
This means ## 3\mid R_{6} ##.
Thus ## R_{6}=111111=3\cdot 7\cdot 11\cdot 13\cdot 37 ##.
Therefore, a product of primes in ## R_{6} ## is ## 3\cdot 7\cdot 11\cdot 13\cdot 37 ##.
Correct.

Or ##111111=111\cdot 1001=(3\cdot 37)\cdot (7\cdot 11 \cdot 13)## also from former results.

Math100
The test with the 3-digit nonalternating sum works for ##111##, according to a Wikipedia page about divisibility rules.

Math100
That's a very clever solution, mine is much simpler:
• ## 3 | 111,111 ## by the sum of digits rule: ## \frac{111,111}{3} = 37,037 ##
• ## \frac{37,037}{37} = 1,001 ## by inspection
• It is well known that ## 11| 10^{(2k + 1)} + 1; \frac{1,001}{11} = 91 ##
• ## 91 = 7 \cdot 13 ## by inspection

## What is a repunit?

A repunit is a number that consists of only repeated digits, such as 111111 or 777777.

## What is the significance of "R6"?

"R6" refers to a specific repunit with 6 digits, in this case 111111.

## Why is it important to factor a repunit into a product of primes?

Factoring a repunit into a product of primes allows us to better understand the number and its properties, and can also be useful in solving certain mathematical problems.

## How do you factor a repunit into a product of primes?

To factor a repunit into a product of primes, you can use methods such as trial division or the Sieve of Eratosthenes to find the prime factors of the number.

## What is the product of primes that make up the repunit R6?

The prime factors of R6 are 3 and 7, so the product of primes that make up R6 is 3 x 7 = 21.

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