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Q1(i):
Express each number as a product of its prime factors: (i) 140
Solution :
Given: The integer $140$.
To find: The prime factorization of $140$, expressed as a product of its prime factors.
Step 1: Understanding Prime Factorization
Prime factorization is the process of expressing a composite number as a product of prime numbers. A prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself. We will use the method of successive division by the smallest prime numbers ($2, 3, 5, 7, 11, \dots$).
Step 2: Sequential Division
We begin by dividing $140$ by the smallest prime number, which is $2$:
$140 \div 2 = 70$ [Since $140$ is an even number, it is divisible by $2$]Next, we divide the quotient $70$ by the smallest prime number, which is $2$:
$70 \div 2 = 35$ [Since $70$ is an even number, it is divisible by $2$]Next, we check if $35$ is divisible by $2$ or $3$. Since $35$ is odd and the sum of its digits ($3+5=8$) is not divisible by $3$, we move to the next prime number, $5$:
$35 \div 5 = 7$ [Since the last digit is $5$, it is divisible by $5$]Finally, we divide the quotient $7$ by the prime number $7$:
$7 \div 7 = 1$ [Since $7$ is a prime number, it is only divisible by itself]Step 3: Compiling the Factors
The prime factors obtained are $2, 2, 5,$ and $7$.
We can write the product as:
$140 = 2 \times 2 \times 5 \times 7$
Step 4: Expressing in Exponential Form
Using the law of exponents, where $a \times a = a^2$:
Final Answer: The prime factorization of $140$ is $2^2 \times 5 \times 7$.
More Questions from Class 10 Mathematics Real numbers EXERCISE 1.1
- Q1(ii): Express each number as a product of its prime factors: (ii) 156
- Q1(iii): Express each number as a product of its prime factors: (iii) 3825
- Q1(iv): Express each number as a product of its prime factors: (iv) 5005
- Q1(v): Express each number as a product of its prime factors: (v) 7429
- Q2(i): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (i) 26 and 91
- Q2(ii): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (ii) 510 and 92
- Q2(iii): Find the LCM and HCF of the following pairs of integers and verify that LCM $\times$ HCF = product of the two numbers. (iii) 336 and 54
- Q3(i): Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21
- Q3(ii): Find the LCM and HCF of the following integers by applying the prime factorisation method. (ii) 17, 23 and 29
- Q3(iii): Find the LCM and HCF of the following integers by applying the prime factorisation method. (iii) 8, 9 and 25
- Q4: Given that HCF (306, 657) = 9, find LCM (306, 657).
- Q5: Check whether $6^n$ can end with the digit 0 for any natural number $n$.
- Q6: Explain why $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
- Q7: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
CBSE Solutions for Class 10 Mathematics Real numbers
Chapters in CBSE - Class 10 Mathematics
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