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.
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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 .
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