What is the smallest number by which 847 must be divided to obtain a perfect square?

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Brandon W.

Algebra

9 months, 3 weeks ago

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find the smallest number by which 2100 must be multiplied so that the product become a perfect square find the square root of the number so obtained

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Video Transcript

I want to find a number that can be used to add up. Break this down into smaller numbers if you lie to get a perfect square and the lowest number. It is 21 times 100 and 7 times 10 square. I must make these two numbers into perfect squares because he is a perfect square. What would happen if I times by 7 and 3? I get 7 square times 3 squared times 10 square, which is a perfect square, and I will get 21 square times 21 that I will get, which is the answer. If I do the square root of 44100 point, I'm going to get 205. You can't find the square root if you don't know the answer to 21 and 210.

Answer

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Hint: To find the required smallest number, we will use the prime factorization method. We will write the given number $ 3388 $ as the multiple of primes. After that it will be written in the form of a group of two primes if possible.

Complete step-by-step answer:
To solve the given problem, we must know the prime factorization method. By using the method of prime factorization, we can express the given number as a product of prime numbers. Therefore, we will write the given number $ 3388 $ as the product of primes. Let us do the prime factorization of $ 3388 $ . Note that $ 3388 $ is an even number so we can start prime factorization with prime number $ 2 $ .

$ 2 $	$ 3388 $
$ 2 $	$ 1694 $
$ 7 $	$ 847 $
$ 11 $	$ 121 $
$ 11 $	$ 11 $
	$ 1 $

Therefore, we can write $ 3388 = 2 \times 2 \times 7 \times 11 \times 11 $ . Here we are dealing with a perfect square so we have to write the obtained factorization in the form of a group of two primes if possible. So, we can write $ 3388 = \left( {2 \times 2} \right) \times 7 \times \left( {11 \times 11} \right) \cdots \cdots \left( 1 \right) $ . To find a perfect square root, we have to take one number from each group of two but here we can see that a single $ 7 $ cannot be written as a group of two.
Let us multiply by $ 7 $ on both sides of the equation $ \left( 1 \right) $ . So, we can write
 $ 3388 \times 7 = \left[ {\left( {2 \times 2} \right) \times 7 \times \left( {11 \times 11} \right)} \right] \times 7 $
 $ \Rightarrow 3388 \times 7 = \left( {2 \times 2} \right) \times \left( {7 \times 7} \right) \times \left( {11 \times 11} \right) $
Here we can take one number from each group of two. So we will get the perfect square root. Hence, the required smallest number is $ 7 $ . Hence, $ 3388 $ should be multiplied by $ 7 $ to be a perfect square number.
Let us take one number from each group of two to find the square root. So, we can write
 $ \sqrt {3388 \times 7} = 2 \times 7 \times 11 = 154 $
Hence, $ 154 $ is square root of the perfect square number.

Note: Remember that if the number is even then it is divisible by $ 2 $ . Double the last digit of the number and subtract the doubled number from the remaining number (remaining digits). If the result is divisible by $ 7 $ then that number is divisible by $ 7 $ . Note that here we will consider positive differences. In the given problem, $ 847 $ is divisible by $ 7 $ because double of last digit $ 7 $ is $ 14 $ and positive difference of remaining number (remaining digit) $ 84 $ and $ 14 $ is $ 70 $ and the number $ 70 $ is divisible by $ 7 $ .

Solution:

(i) 252 = 2 x 2 x 3 x 3 x 7

Here, prime factor 7 has no pair. Therefore 252 must be divided by 7 to make it a perfect square.

\therefore252\div7=36

And \sqrt{36}=2\times3=6

(ii) 2925 = 3 x 3 x 5 x 5 x 13

Here, prime factor 13 has no pair. Therefore 2925 must be divided by 13 to make it a perfect square.

\therefore2925\div13=225

And \sqrt{225}=3\times5=15

(iii) 396 = 2 x 2 x 3 x 3 x 11

Here, prime factor 11 has no pair. Therefore 396 must be divided by 11 to make it a perfect square.

\therefore396\div11=36

And \sqrt{36}=2\times3=6

(iv) 2645 = 5 x 23 x 23

Here, prime factor 5 has no pair. Therefore 2645 must be divided by 5 to make it a perfect square.

\therefore2645\div5=529

And \sqrt{529}=23

(v) 2800 = 2 x 2 x 2 x 2 x 5 x 5 x 7

Here, prime factor 7 has no pair. Therefore 2800 must be divided by 7 to make it a perfect square.

\therefore2800\div7=400

And \sqrt{400}=2\times2\times5=20

(vi) 1620 = 2 x 2 x 3 x 3 x 3 x 3 x 5

Here, prime factor 5 has no pair. Therefore 1620 must be divided by 5 to make it a perfect square.

\therefore1620\div5=324

And \sqrt{324}=2\times3\times3=18

Video transcript

hello, students welcome to Lido homework in this video we are going to solve this question that us asks us to find the smallest whole number that needs to be divided um from the numbers to get a perfect square so we are going to have to divide a certain number so we get a perfect square and once we get that perfect square we also have to find out the square root so let's get into the first one is 252. okay so let's prime factorize this and see what the factors we are getting and which one we need to eliminate okay so 252 goes in by 2 so you get 126 2 6 are 2 3's are this goes in by 3 21 3 7's are and seven ones are okay so a quick uh revision over here the question can also be asked is the smallest number whole number you need to multiply okay and when we do that we need we try to complete the groups that uh we have received over here so for example in this question the factors i have gotten is 2 into 2 into 3 into 3 into 7 okay so in case they ask you multiply that means out of the groups that are formed i have a group of 2 i have a group of 3 and 7. so 7 is a single number it's not a complete group it's not 7 into 7 so you will have to multiply 7 right but this question says which number you should divide so you need to eliminate numbers that do not have complete group so over here obvious the number that you need to eliminate is seven so you have proper groups of two and three complete groups of two and three right so the number you need to eliminate here is seven so we'll divide 252 by 7. when you divide that by 7 you get the answer as 36 so 36 is a perfect square and the second part of the question asks us to find the square root of 36 we know if you practice enough by now you'll be able to identify that 6 6 are 36 but still if you cannot remember that take the square root sign write the other factors are there in the square root sign over here 2 into 2 3 into 3 right take out each one from each of the groups and when you do that you don't write the square root sign so one two from the first group and a three from the second group two threes are six so the square root of 36 is six okay let's solve the second one-two nine two five now this will go in by three so this is nine seven 5 again this will go in by 3 3 2 3 5's are 15 okay now 325 will be divisible by 5. so five-six are thirty-five fives are Twenty-five five ones are five threes are okay thirteen is a prime number so it will just go in with itself thirteen ones are 13 just erase these okay 13 ones are 13 and that's the end of the prime factorization let's write down all the factors that we've received here 3 into 5 into 13 now let's group and see which number needs to be eliminated that is 3 has a group 5 has a group 13 is just single so I'll have to eliminate 13 and to eliminate 13 is to divide 2 9 to 5 by 13 this will give you 225 so the perfect square is 225 let's find out the square root of 225 so square root we'll take in the available groups 3 into 5 pair them up and take each number from each group so 3 into 5 and that's 15 so the square root of 225 is 15. Let's move on to the third one, the third one says 396. so lets the prime factor this out and see what the factors that we're getting 2 198 2 9 18 2 9 so 18 99 this will go into three threes are again this will go into three. and we know 11 goes in by itself, so we've got the prime factors here, so that is 2 into 3 into 11. look at the number that doesn't have a complete group, 2 has a group here, 3 has a group here, 11 is a single number so that means we must divide 396 by 11 to get a perfect square, so 396 divided by 11 is 36. so the square root of 36 like we saw in the first question, we know it's 6 by now so you can either write 6 but you have to show the steps write the complete step as 2 into 3 we've eliminated 11 and now we'll take each digit from each of the groups and multiply them and we'll get the answer as 6. Okay, the square root of 36 is 6. Let's move on to the fourth one that says 2 6 4 5 okay so let's prime factorize this goes in by 5 529 okay 529 is a perfect square of 23. so 23 23s are 529 and 23 ones are so the factors that you've got here are 5 into 23 into23 now clearly there's just one group here that's 23 into 23 and 5 is a single-digit so that means I must divide 2 6 4 5 by 5 because I need to eliminate 5 so I'll divide it by 5 which will give me 529 and like you see in the prime factorization step here 529 is a perfect square of 23 the square root of 529 is equal to if you express it we'll express it like this 23 into 23 and we just take out one number so the square root of 529 is 23. let's move on to the fifth one that says 2 8 double zero okay this is 0 so it will go in by two one four double zero two sevens are fourteen zero moves ahead two ones are two sevens are fourteen carries one-two fives are ten okay 175 this will go in by five threes are and five fives are five sevens are and seven ones are okay let's write the factors 2 into 5 into 7 okay let's group them and see which number we need to eliminate or divide 2 has a group of 5 as a group 7 is a single number so I must divide two eight double zeros by seven. I'll get the answer as 400 right so the square root of 400 will be the square root of all the factors eliminating 7 so it will be 2 into 5. you have 3 groups here that take each number out of each of the group 2 into 5 which is 20. so square root of 400 is 20. the last number here is one six two zero let's solve for this so two eights are two ones are zero twos are zero and five two fours are eight sorry two fours are eight zero and five three once threes and fives are here. three fours are twelve three fives are fifteen, okay forty-five will again go into three ones are three fives are fifteen three fives are and five ones are okay so over here the next step is to just write all the factors so let's write the factors we have 2 into 3 into 3 into 5. let's group them up and see which number we need to eliminate so 2 has a group, 3 has a group, 5 is single, so we must eliminate 5 that is divide 1 6 2 0 by 5. You'll get the answer as 324 so the square root of 324 will be a square root of 2 into 3 into 3 okay take each digit from each group 2 into 3 which is nothing but 18 so the square root of 324 is 18. Hope I could uh solve your doubts about this question that's all in this video please feel free to share and comment and I'll see you in the next video thank you

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