Question 1207226: 3-digit number to be the
largest prime number less than 1000. What is this
number?
Found 3 solutions by ikleyn, math_tutor2020, greenestamps:
Answer by ikleyn(52803)   (Show Source):
You can put this solution on YOUR website! .
In the Internet, open a table of prime numbers, for example, this one
https://www.mathforblondes.com/2023/10/table-of-primes.html#google_vignette
or this
https://en.wikipedia.org/wiki/List_of_prime_numbers
Find there the greatest 3-digit number less than 1000.
It is 997. ANSWER
Solved.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: 997
Explanation
One method is to look at a list of primes. There's not much else to say about this method.
Another method would be to start at 999 and count down until finding the prime we're after.
We see that 999 is not prime because the digits add to a multiple of 3, so 3 is a factor.
998 isn't prime either since it is even (2 is a factor).
Now to check 997.
This number isn't even, so we rule 2 out as a factor. The digits do not add to a multiple of 3, so that is ruled out as well.
sqrt(997) = 31.575 approximately
We need to check all primes up to 31. Those primes are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
This is a fairly short list to work with. You should find that none of these values are a factor of 997, which lets us conclude that 997 is prime. It's the largest prime smaller than 1000.
One tool that can be used to confirm is WolframAlpha
https://www.wolframalpha.com/input?i=largest+prime+number+less+than+1000
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Let's suppose this question is on a timed test, where you can't look at a list of primes on the internet. Then your task is to consider odd numbers close to 1000 and test to see if the number is divisible by any smaller prime numbers.
A possible (but not absolutely necessary) first step is to note that the square root of 1000 is between 31 and 32, so 31 is the largest prime you will need to check. So the prime numbers we need to look at are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.
Note that when we are performing calculations to see if our number is divisible by a smaller prime, we are not looking for the quotient -- we are only testing for divisibility. We don't want to do actual long division; that is too slow.
When we are testing to see if our large odd 3-digit integer n is divisible by a prime p, there are two steps we can perform (repeatedly, if necessary) to make it faster and easier to check for divisibility:
(1) Add or subtract any multiple of p to n. If n is divisible by p, then n plus or minus any multiple of p will be divisible by p
(2) Knock off any trailing zeros. If 10 times a number is divisible by p, then the number is divisible by p.
So to test for divisibility by p, we add or subtract a multiple of p to our number so that the new number has a trailing zero, and then we knock off that trailing zero.
Let's use this process to find the largest 3-digit prime number.
999 is clearly not prime, because it is divisible by 3. So next we look at 997.
See if 997 is divisible by 7: subtract 7 from 997 to get 990 and knock off the trailing 0 to get 99. 99 is not divisible by 7, so neither is 997.
See if 997 is divisible by 11: Add 33 to 997 to get 1030 and knock off the trailing 0 to get 103. 103 is not divisible by 11, so neither is 997.
See if 997 is divisible by 13: Add 13 to 997 to get 1010 and knock off the trailing 0 to get 101. 101 is not divisible by 13, so neither is 997.
See if 997 is divisible by 17: Subtract 17 from 997 to get 980 and knock off the trailing 0 to get 98. 98 is not divisible by 17, so neither is 997.
See if 997 is divisible by 23: Add 23 to 997 to get 1020 and knock off the trailing 0 to get 102. 102 is not divisible by 23, so neither is 997.
See if 997 is divisible by 29: Subtract 3 times 29 (=87) from 997 to get 910 and knock off the trailing 0 to get 91. 91 is not divisible by 29, so neither is 997.
See if 997 is divisible by 31: Add 3 times 31 (=93) to 997 to get 1090 and knock off the trailing 0 to get 109. 109 is not divisible by 31, so neither is 997.
We have tested 997 for divisibility by all the primes less than or equal to 31 and found that 997 is not divisible by any of them. That means 997 is prime.
ANSWER: 997
That looks like a lot of work; but it is not too bad if your mental arithmetic is good.
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