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In our last post, we discussed a method to identify prime numbers and also posted a question where you were required to apply these concepts in solving the question. If you have not seen our earlier posts, here are the links for you to have a look:
Part 1 -https://bit.ly/2Yu4JMY
Part 2 - https://bit.ly/2WPWMS6
Today, we will start off by solving the problem that we posted with yesterday’s post. Post this, we will look at some important areas where Prime numbers are applied.
So, let’s get started.
What is the highest prime number that can divide 10! * 11! + 11! * 12! ?
A. 17
B. 7
C. 19
D. 21
E. 23
The first step in solving this question, is to simplify the expression given. Since 11! is common in both the terms, we can take 11! as common. Doing this, we get, 11! ( 10! + 12!).
In the terms inside the braces, 10! is common; so (10! + 12!) can be simplified as [10! (1 + 12 * 11) ].
Therefore, the given expression can be written as 10! * 11! (133). Now, if we are to find the highest prime factor that can divide the given expression, we need to find out the highest prime factor which can divide 133. But before that, we need to figure out if 133 itself is prime.
Look at the options. If 133 were a prime number, it was sure to find a place in the options. 133 not being one of the options is a clear clue that the question is giving you that 133 is not prime.
Let us apply the method of identifying primes to confirm this. The nearest perfect square to 133 is 121. The square root of 121 is 11. Prime numbers lesser than or equal to 11 are 2,3,5,7 and 11.
Clearly, 133 is not divisible by 2 and 5. The sum of the digits of 133 is 7, therefore it is not divisible by 3 either. But, you will see that 133 is divisible by 7, since 133 = 7 * 19.
Therefore, 133 is not a prime number. Also, the biggest prime number that can divide the given expression is 19. Therefore, the correct answer option is C.
When checking for primes, apart from the approach that we described in our previous post, you can always check for divisibility by smaller numbers like 3, 9 and 11, because these numbers have very simple divisibility rules. More often than not, on the GMAT, you will see that the values will be designed in such a way that you will be able to eliminate non-primes if you check for divisibility by some of these numbers.
Now, let us look at some of the areas in which you can expect questions related to Prime numbers. One of the common types, obviously, is to test you on the basic property of a prime number, like how you saw in the problem above. But, probably the most important application of Prime numbers is in Prime factorization.
Prime factorization, as you know, is a process of breaking down composite numbers into their constituent prime factors. This process also forms the basis for concepts like HCF and LCM, factors of composite numbers and also factorial questions. In all these topics, you will be required to be familiar with prime number concepts. In all of the topics mentioned above, you will be required to prime factorise 2-digit numbers or fairly large 3-digit numbers. In cases of large 3-digit numbers, instead of prime factorizing the normal way, adopt a smart approach by doing something like this:
Let’s say we have to prime factorise 750. Instead of prime factorizing like how we usually do, we can say 750 = 75 * 10. Now, 75 = 25 * 3 and 25 = \(5^2\); therefore, 75 = \(5^2\) * 3 and of course, 10 = 5 * 2.
Therefore, 750 = \(5^3 * 3 * 2.\)
This way, you will not only save time and energy, but it will also make you feel confident of dealing with larger numbers without getting overwhelmed by their magnitude.
Of course, since all areas of Math are inter-related in a lot of ways, you will be expected to apply your knowledge of Prime numbers in questions on Algebra or Statistics or Geometry. One area in Geometry where you will probably deal with Prime numbers more extensively than others, is Pythagorean triplets. All primitive Pythagorean triplets will always have a prime number as part of the triplets.
Having a sound knowledge of prime numbers will help you in solving questions in a lot of topics in Math. If this was an area where you faced some trouble, we hope that our posts on Prime numbers have been of help to you.
In tomorrow’s post, we will quickly summarise everything related to this discussion on Prime numbers, so that you have a quick ready reckoner of sorts on Prime numbers. We will also post some really good questions based on Prime numbers.
Thank you!
Part 1 -https://bit.ly/2Yu4JMY
Part 2 - https://bit.ly/2WPWMS6
Today, we will start off by solving the problem that we posted with yesterday’s post. Post this, we will look at some important areas where Prime numbers are applied.
So, let’s get started.
What is the highest prime number that can divide 10! * 11! + 11! * 12! ?
A. 17
B. 7
C. 19
D. 21
E. 23
The first step in solving this question, is to simplify the expression given. Since 11! is common in both the terms, we can take 11! as common. Doing this, we get, 11! ( 10! + 12!).
In the terms inside the braces, 10! is common; so (10! + 12!) can be simplified as [10! (1 + 12 * 11) ].
Therefore, the given expression can be written as 10! * 11! (133). Now, if we are to find the highest prime factor that can divide the given expression, we need to find out the highest prime factor which can divide 133. But before that, we need to figure out if 133 itself is prime.
Look at the options. If 133 were a prime number, it was sure to find a place in the options. 133 not being one of the options is a clear clue that the question is giving you that 133 is not prime.
Let us apply the method of identifying primes to confirm this. The nearest perfect square to 133 is 121. The square root of 121 is 11. Prime numbers lesser than or equal to 11 are 2,3,5,7 and 11.
Clearly, 133 is not divisible by 2 and 5. The sum of the digits of 133 is 7, therefore it is not divisible by 3 either. But, you will see that 133 is divisible by 7, since 133 = 7 * 19.
Therefore, 133 is not a prime number. Also, the biggest prime number that can divide the given expression is 19. Therefore, the correct answer option is C.
When checking for primes, apart from the approach that we described in our previous post, you can always check for divisibility by smaller numbers like 3, 9 and 11, because these numbers have very simple divisibility rules. More often than not, on the GMAT, you will see that the values will be designed in such a way that you will be able to eliminate non-primes if you check for divisibility by some of these numbers.
Now, let us look at some of the areas in which you can expect questions related to Prime numbers. One of the common types, obviously, is to test you on the basic property of a prime number, like how you saw in the problem above. But, probably the most important application of Prime numbers is in Prime factorization.
Prime factorization, as you know, is a process of breaking down composite numbers into their constituent prime factors. This process also forms the basis for concepts like HCF and LCM, factors of composite numbers and also factorial questions. In all these topics, you will be required to be familiar with prime number concepts. In all of the topics mentioned above, you will be required to prime factorise 2-digit numbers or fairly large 3-digit numbers. In cases of large 3-digit numbers, instead of prime factorizing the normal way, adopt a smart approach by doing something like this:
Let’s say we have to prime factorise 750. Instead of prime factorizing like how we usually do, we can say 750 = 75 * 10. Now, 75 = 25 * 3 and 25 = \(5^2\); therefore, 75 = \(5^2\) * 3 and of course, 10 = 5 * 2.
Therefore, 750 = \(5^3 * 3 * 2.\)
This way, you will not only save time and energy, but it will also make you feel confident of dealing with larger numbers without getting overwhelmed by their magnitude.
Of course, since all areas of Math are inter-related in a lot of ways, you will be expected to apply your knowledge of Prime numbers in questions on Algebra or Statistics or Geometry. One area in Geometry where you will probably deal with Prime numbers more extensively than others, is Pythagorean triplets. All primitive Pythagorean triplets will always have a prime number as part of the triplets.
Having a sound knowledge of prime numbers will help you in solving questions in a lot of topics in Math. If this was an area where you faced some trouble, we hope that our posts on Prime numbers have been of help to you.
In tomorrow’s post, we will quickly summarise everything related to this discussion on Prime numbers, so that you have a quick ready reckoner of sorts on Prime numbers. We will also post some really good questions based on Prime numbers.
Thank you!
01 Jun 2024, 02:31
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Hello folks,
This was yet another brilliant post about prime numbers.
The clear explanation helped me a great deal with understanding primes better. I recommend everyone struggling with this topic to check out this post.
Study smart!
This was yet another brilliant post about prime numbers.
The clear explanation helped me a great deal with understanding primes better. I recommend everyone struggling with this topic to check out this post.
Study smart!
