The foundation of every mathematical concept depends on mastering numerical principles and their attributes. The fundamental principle in mathematics demonstrates how to break numbers into prime number factors to understand their internal makeup for better calculation purposes. Prime factorisation serves as a valuable method that reveals how to find both common divisors and multiples. The technique establishes a solid foundation to recognize numerical connections between numbers and functions effectively with extensive calculations and real-world applications about time duration and quantitative and repetitive scenarios.
This Story also Contains
The exercise provided through NCERT Solutions for Class 10 Maths in the standard NCERT Books teaches students about prime factorization in combination with the Highest Common Factor and the Lowest Common Multiple. All concepts covered in this exercise become vital building blocks for both arithmetic and algebra. The instructional materials teach students two methods for solving complex problems and confirming procedural validity. The questions that use digits with factorials alongside divisibility tests help students to develop their logical thinking abilities and strengthen their grasp of real numbers and their characteristics.
Q1 (1) Express each number as a product of its prime factors: 140
Answer:
By taking LCM of the number 140 we can express it as a product of its prime factors, like as follows:
$140=2\times 2\times 5\times 7$
$140=2^{2}\times 5\times 7$
Q 1 (2) Express each number as a product of its prime factors: 156
Answer:
By taking LCM of the number 156 we can express it as a product of its prime factors, like as follows:
$156=2\times 2\times 3\times 13$
$156=2^{2}\times 3\times 13$
Q1 (3) Express each number as a product of its prime factors: 3825
Answer:
By taking LCM of the number 3825 we can express it as a product of its prime factors, like as follows:
$3825=3\times 3\times 5\times 5\times 17$
$3825=3^{2}\times 5^{2}\times 17$
Q1 (4) Express each number as a product of its prime factors: 5005
Answer:
By taking LCM of the number 5005 we can express it as a product of its prime factors, like as follows:
$5005=5\times 7\times 11\times 13$
Q1 (5) Express each number as a product of its prime factors: 7429
Answer:
By taking LCM of the number 7429 we can express it as a product of its prime factors, like as follows:
$7429=17\times 19\times 23$
Answer:
Express the numbers as the product of their prime numbers.
26 = 2 × 13
91 = 7 × 13
HCF(26, 91) = 13
LCM(26, 91) = 2 × 7 × 13 = 182
Verification; HCF × LCM = 13 × 182 = 2366
26 × 91 = 2366
26 × 91 = HCF × LCM
Hence, verified.
Answer:
The number can be expressed as the product of prime factors as
510 = 2 × 3 x 5 × 17
92 = 22 × 23
HCF (510, 92) = 2
LCM (510, 92) = 22 × 3 x 5 × 17 × 23 = 23460
Verification; HCF x LCM = 2 × 23460 = 46920
510 × 92 = 46920
510 × 92 = HCF × LCM
Hence, verified
Answer:
Express the numbers as the product of their prime numbers.
336 = 24 × 3 × 7
54 = 2 × 33
HCF(336,54) = 2 x 3 = 6
LCM(336,54) = 24 × 33 × 7 = 3024
Verification; HCF × LCM = 6 × 3024 = 18144
336 × 54 = 18144
336 × 54 = HCF × LCM
Hence, verified
Answer:
The numbers can be written as the product of their prime factors as follows
12 = 22 x 3
15 = 3 x 5
21 = 3 x 7
HCF = 3
LCM = 22 x 3 x 5 x 7 = 420
Answer:
The given numbers are written as the product of their prime factors as follows
17 = 1 × 17
23 = 1 × 23
29 = 1 × 29
HCF = 1
LCM = 17 × 23 × 29 = 11339
Answer:
The given numbers are written as the product of their prime factors as follows
8 = 23
9 = 32
25 = 52
HCF = 1
LCM = 23 × 32 × 52 = 1800
Q4 Given that HCF (306, 657) = 9, find LCM (306, 657).
Answer:
We know: HCF × LCM = a × b
So, HCF (306, 657) × LCM (306, 657) = 306 × 657
$LCM (306, 657) = \frac{306\times 657}{HCF (306, 657) }$
$ LCM (306, 657) = \frac{306\times 657}{9}$
$LCM (306, 657) =22338$
Q5 Check whether $6 ^n$ can end with the digit 0 for any natural number n.
Answer:
By prime factorisation, we have
6n = 2n × 3n
If a number's prime factorisation includes at least 1 as the power of both 2 and 5, then the number will conclude with 0. We can infer that, for every value of n, 6n will terminate with the number 0 since the prime factorisation of 6n shows that the power of 5 equals 0.
Answer:
7 × 11 × 13 + 13
= (7 × 11 + 1) × 13
= 78 × 13 or 2 × 3 × 132
Similarly, 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
= (7 × 6 × 4 × 3 × 2 × 1 + 1) × 5
= 5 × 1008
The number rule states that we can take at least two common numbers out of two, and after solving, we saw that both numbers were even, as a result of which the number is composite.
Answer:
The time after which they meet again at the starting point will be equal to the LCM of the times they individually take to complete one round.
Time taken by Sonia = 18 = 2 × 32
Time taken by Ravi = 12 = 22 × 3
LCM (18, 12) = 22 × 32 = 36
Therefore, they would again meet at the starting point after 36 minutes.
1. Prime Factorization: Individuals learn the process to decompose whole numbers into one-of-a-kind factors consisting of prime numbers. Students need prime factorization as the base for solving multiple mathematical problems that include fraction simplification and LCM and HCF problem-solving and number property analysis in arithmetic and algebra.
Also Read:
Also see:
Students must check the NCERT solutions for class 10 of the Mathematics and Science Subjects.
Students must check the NCERT Exemplar solutions for class 10 of the Mathematics and Science Subjects.
Frequently Asked Questions (FAQs)
Theorem: Every composite number may be factored as a product of primes, which means that any composite number can be expressed as a product of primes
The questions are based on the concept of the Fundamental Theorem of Arithmetic, in finding HCF and LCM, questions related to the relationship of HCF and LCM.
The prime factors of 144,
144 = 2 × 2 × 2 × 2 × 3 × 3
Students can practice problems discussed in class 10 ex 1.2 to command these concepts.
Zero is neither prime nor a composite number because it does not have any prime factors.
The full form of HCF is “Highest Common Factor” and LCM is “Least common multiple”.
The methods used in finding LCM and HCF are the prime factorization method and long division method.
104 = 2 × 2 × 2 × 13
160 = 2 × 2 × 2 × 2 × 2 × 5
The common factors of 104 and 160 are 2 × 2 × 2 = 8
Therefore, HCF (104, 160) = 8
21=3×7
28=4×7
Now, LCM (21, 28)=7×4×3=84
The formula that involves both HCF and LCM is
(HCF of the two numbers) x (LCM of the two numbers)= Product of Two numbers
Let the other number be a.
We know that,
(HCF of the two numbers) x (LCM of the two numbers)= Product of Two numbers
Now, 7×84=21×a
588=21a
Which on dividing both sides by 21 ,
We get, a=28
The other number is 28 .
On Question asked by student community
To get the previous year question papers you can visit the official website of your board and search under the section of previous year question papers. You can also search on Google for the papers or visit the educational platforms like Careers 360 for the papers. They also provide with the papers and answer key also.
You must be at least 14 years old by December 31st of the year 2027
Since your date of birth is 29 January 2013, you will turn 14 in January 2027, which is before the December 31st deadline for the 2027 exam.
Hence you are eligible..
Good luck!!
Hello,
If you want to get your 10th marksheet online, you just need to visit an official website like https://www.cbse.gov.in/ or https://results.cbse.nic.in/ for the CBSE board, and for the state board, you can check their website and provide your roll number, security PIN provided by the school, and school code to download the result.
I hope it will clear your query!!
Hello, if you are searching for Class 10 books for exam preparation, the right study material can make a big difference. Standard textbooks recommended by the board should be your first priority as they cover the syllabus completely. Along with that, reference books and guides can help in practicing extra questions and understanding concepts in detail. You can check the recommended books for exam preparation from the link I am sharing here.
https://school.careers360.com/ncert/ncert-books-for-class-10
https://school.careers360.com/boards/cbse/cbse-best-reference-books-for-cbse-class-10-exam
Hello Dinesh !
As per CBSE board guidelines for internal assessment for class 10th you will have to give a 80 marks board exam and 20 marks internal assessment. The internal assessment will be at the end of your year.
For knowing the definite structure of the internal assessment you will have to ask your teachers or your seniors in the school as CBSE has provided flexibility in choosing the methods of internal assessment to schools. For more details related to assessment scheme for class 10 given by CBSE you can visit: Assessment scheme (http://cbseacademic.nic.in/web_material/CurriculumMain2Sec/Curriculum_Sec_2021- 22.pdf)
I Hope you have understood it!
This ebook serves as a valuable study guide for NEET 2025 exam.
This e-book offers NEET PYQ and serves as an indispensable NEET study material.
As per latest syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
As per latest syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters
As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE
As per latest syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters