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15 May 2019, 02:30
Kudos
Bookmarks
Hello,
Greetings for the day!
In yesterday’s post, we discussed the basic properties of Prime Numbers and a couple of methods to identify them. To view the post, please click on this link:
https://bit.ly/2Yu4JMY
But, both these methods had their own limitations in being able to help us in identifying a prime number.
We had mentioned in our post yesterday that we will be discussing a foolproof approach to identifying Prime numbers. Therefore, today, let’s look at a method that can be applied on any number to ascertain whether it is prime or not.
Let’s suppose we are trying to ascertain if a number X is prime or not. Using a very simple algorithm, involving a few simple steps, you will be able to find out if X is prime. So, what’s this algorithm? We have listed it out in a step by step manner so that it becomes easy for you to perform the calculations:
First – Find out the nearest perfect square to X, lesser than X.
Second – Calculate the square root of this perfect square that you just found out in the previous step.
Third – List out all prime numbers which are lesser or equal to the square root you calculated in the second step.
Fourth – Check whether the given number X is divisible by any of the prime numbers listed down by you in the third step.
Fifth – If it is divisible by even ONE of the prime numbers, then it means X has more than 2 factors (remember the fact that any positive integer will definitely have 2 factors – 1 and itself) and hence is NOT prime.
On the other hand, if X is not divisible by any of the prime numbers listed down by you, then X is a prime.
This method can be applied to any number and does not have exceptions like the previous two methods. The only pre-requisites, for you to apply this method effectively, are that you need to know basic divisibility rules and at least the first 15 prime numbers by heart.
Having said that, this is not to say that the methods discussed in our previous post, are totally ineffective. In a lot of questions where you need to identify prime numbers, we would still recommend you to start off by using the (6k) - 1 and the (6k) + 1 concept, to eliminate at least 1 or 2 options. Post this, you can use the method that we described today to zero in on the Prime number, of the remaining options.
In the third part of this post, tomorrow, we will look at a few important application areas of Prime numbers so that you get a hang of how to apply the concepts that we have discussed till now.
Greetings for the day!
In yesterday’s post, we discussed the basic properties of Prime Numbers and a couple of methods to identify them. To view the post, please click on this link:
https://bit.ly/2Yu4JMY
But, both these methods had their own limitations in being able to help us in identifying a prime number.
We had mentioned in our post yesterday that we will be discussing a foolproof approach to identifying Prime numbers. Therefore, today, let’s look at a method that can be applied on any number to ascertain whether it is prime or not.
Let’s suppose we are trying to ascertain if a number X is prime or not. Using a very simple algorithm, involving a few simple steps, you will be able to find out if X is prime. So, what’s this algorithm? We have listed it out in a step by step manner so that it becomes easy for you to perform the calculations:
First – Find out the nearest perfect square to X, lesser than X.
Second – Calculate the square root of this perfect square that you just found out in the previous step.
Third – List out all prime numbers which are lesser or equal to the square root you calculated in the second step.
Fourth – Check whether the given number X is divisible by any of the prime numbers listed down by you in the third step.
Fifth – If it is divisible by even ONE of the prime numbers, then it means X has more than 2 factors (remember the fact that any positive integer will definitely have 2 factors – 1 and itself) and hence is NOT prime.
On the other hand, if X is not divisible by any of the prime numbers listed down by you, then X is a prime.
This method can be applied to any number and does not have exceptions like the previous two methods. The only pre-requisites, for you to apply this method effectively, are that you need to know basic divisibility rules and at least the first 15 prime numbers by heart.
Having said that, this is not to say that the methods discussed in our previous post, are totally ineffective. In a lot of questions where you need to identify prime numbers, we would still recommend you to start off by using the (6k) - 1 and the (6k) + 1 concept, to eliminate at least 1 or 2 options. Post this, you can use the method that we described today to zero in on the Prime number, of the remaining options.
In the third part of this post, tomorrow, we will look at a few important application areas of Prime numbers so that you get a hang of how to apply the concepts that we have discussed till now.
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15 May 2019, 02:36
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Here's a question for all of you to apply the techniques discussed in our last 2 posts:
What is the highest prime number that can divide 10! * 11! + 11! * 12! ?
A. 17
B. 7
C. 19
D. 21
E. 23
Answers for this question will be posted along with tomorrow's post on this topic.
Happy Solving
What is the highest prime number that can divide 10! * 11! + 11! * 12! ?
A. 17
B. 7
C. 19
D. 21
E. 23
Answers for this question will be posted along with tomorrow's post on this topic.
Happy Solving
15 May 2019, 08:50
Kudos
Bookmarks
ArvindCrackVerbal
Checking if a number is in the form 6k-1 or 6k+1 amounts to checking that the number is divisible neither by 2 nor by 3, which you can do almost instantly using divisibility tests.
ArvindCrackVerbal
In case this might mislead test takers about what the GMAT might ask them to do: you would never need to use all of the first fifteen primes in a prime-testing situation on the GMAT. That is, you will never see a number roughly equal to 47^2 and need to prove that it's prime, because that would take far too long. It cannot be true in a GMAT question that you need to figure out that, say, 2213 is prime, because the fastest way to prove that is by using the method Arvind describes, and that method involves checking if 2213 is divisible by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47 (after first working out that 47^2 is the nearest perfect square just less than 2213). Almost no one can do that in two minutes. If there were a number roughly that big in a GMAT question, say 2107 or 2199, and for some reason it was important to decide whether the number was prime, the answer will simply always be "it is not prime", because they could never ask you in two minutes to prove that it is. The number will have some obvious divisor: 2107 is divisible by 7 (because 2107 = 2100 + 7 = 7(301)), for example, or 2199 is divisible by 3 (which we can see immediately by summing its digits).
