Determine whether the integer 701 is prime by testing?

In summary, determining whether an integer is prime means checking if it is only divisible by 1 and itself. This can be done using trial division, where the number is divided by every integer from 2 to its square root. It is important to determine primality for various applications in mathematics and computer science. There are faster methods, such as the Sieve of Eratosthenes and Miller-Rabin test. While all integers can be tested for primality, larger numbers may take longer to test and some extremely large numbers may currently be impossible to test due to computing limitations.
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Homework Statement
Determine whether the integer ## 701 ## is prime by testing all primes ## p\leq\sqrt{701} ## as possible divisors. Do the same for the integer ## 1009 ##.
Relevant Equations
None.
Proof:

Consider all primes ## p\leq\sqrt{701}\leq 27 ##.
Note that ## 701=2(350)+1 ##
## =3(233)+2 ##
## =5(140)+1 ##
## =7(100)+1 ##
## =11(63)+8 ##
## =13(53)+12 ##
## =17(41)+4 ##
## =19(36)+17 ##
## =23(30)+11 ##.
Thus, no prime numbers less than ## 27 ## are divisible by the integer ## 701 ##.
Therefore, the integer ## 701 ## is prime.
Now, we consider all primes ## p\leq\sqrt{1009}\leq 32 ##.
Note that ## 1009=2(504)+1 ##
## =3(336)+1 ##
## =5(201)+4 ##
## =7(144)+1 ##
## =11(91)+8 ##
## =13(77)+8 ##
## =17(59)+6 ##
## =19(53)+2 ##
## =23(43)+20 ##
## =29(34)+23 ##
## =31(32)+17 ##.
Thus, no prime numbers less than ## 32 ## are divisible by the integer ## 1009 ##.
Therefore, the integer ## 1009 ## is prime.
 
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I didn't check all the numbers but it looks like you're doing the right thing
 
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FAQ: Determine whether the integer 701 is prime by testing?

1. How do you test if a number is prime?

The most common method for testing if a number is prime is by using the "divisibility test". This involves dividing the number by all integers from 2 to the square root of the number. If the number is evenly divisible by any of these integers, then it is not prime.

2. What is the square root of 701?

The square root of 701 is approximately 26.4575131106.

3. Can you use a calculator to determine if a number is prime?

Yes, you can use a calculator to determine if a number is prime. However, this method is not always accurate and it is recommended to use the divisibility test for more reliable results.

4. What is the smallest prime number?

The smallest prime number is 2. This is because prime numbers are defined as any number that is only divisible by 1 and itself, and 2 is the smallest positive integer that meets this criteria.

5. Is 701 a prime number?

Yes, 701 is a prime number. It is only divisible by 1 and itself, and it is not evenly divisible by any other integer.

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