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

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.
  • #1
Math100
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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 ##.
 
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  • #2
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.
 
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  • #3
The test with the 3-digit nonalternating sum works for ##111##, according to a Wikipedia page about divisibility rules.
 
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  • #4
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
 

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

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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