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Q1(v):
Express each number as a product of its prime factors: (v) 7429
Solution :
Given: The integer $7429$.
To Find: The prime factorization of $7429$.
Step 1: Testing for divisibility by small prime numbers.
We test the divisibility of $7429$ by the sequence of prime numbers: $2, 3, 5, 7, 11, 13, 17, \dots$
1. Divisibility by 2: The last digit is $9$ (odd), so it is not divisible by $2$.
2. Divisibility by 3: The sum of the digits is $7 + 4 + 2 + 9 = 22$. Since $22$ is not divisible by $3$, $7429$ is not divisible by $3$.
3. Divisibility by 5: The last digit is not $0$ or $5$, so it is not divisible by $5$.
4. Divisibility by 7: $7429 \div 7 = 1061$ with a remainder of $2$. Not divisible.
5. Divisibility by 11: The alternating sum of digits is $7 - 4 + 2 - 9 = -4$. Not divisible by $11$.
6. Divisibility by 13: $7429 \div 13 = 571$ with a remainder of $6$. Not divisible.
7. Divisibility by 17: $7429 \div 17 = 437$. Since the remainder is $0$, $17$ is a prime factor.
Step 2: Factoring the quotient $437$.
We continue testing for divisibility starting from $17$ (since prime factors are non-decreasing):
1. Divisibility by 17: $437 \div 17 = 25$ with a remainder of $12$. Not divisible.
2. Divisibility by 19: $437 \div 19 = 23$. Since the remainder is $0$, $19$ is a prime factor.
Step 3: Factoring the quotient $23$.
Since $23$ is a prime number, it is only divisible by $1$ and itself.
Step 4: Compiling the prime factors.
The prime factorization is obtained by multiplying the prime divisors found in the previous steps:
$7429 = 17 \times 437$
$7429 = 17 \times 19 \times 23$
[Justification: By the Fundamental Theorem of Arithmetic, every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.]
Final Answer: 17 \times 19 \times 23
More Questions from Class 10 Mathematics Real numbers EXERCISE 1.1
- Q1(i): Express each number as a product of its prime factors: (i) 140
- 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
- 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
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