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Q1(iii):
Express each number as a product of its prime factors: (iii) 3825
Solution :
Given: The number $3825$.
To Find: The prime factorization of $3825$.
Step 1: Testing for divisibility by the smallest prime numbers.
We begin by testing the divisibility of $3825$ using the rules of divisibility for prime numbers in ascending order ($2, 3, 5, 7, 11, \dots$).
Check for $2$: The last digit of $3825$ is $5$, which is odd. Therefore, $3825$ is not divisible by $2$.
Check for $3$: The sum of the digits of $3825$ is $3 + 8 + 2 + 5 = 18$. Since $18$ is divisible by $3$, the number $3825$ is divisible by $3$.
$3825 \div 3 = 1275$
Step 2: Continuing the factorization of the quotient.
Now, we factorize $1275$.
Check for $3$: The sum of the digits of $1275$ is $1 + 2 + 7 + 5 = 15$. Since $15$ is divisible by $3$, $1275$ is divisible by $3$.
$1275 \div 3 = 425$
Step 3: Continuing the factorization of the new quotient.
Now, we factorize $425$.
Check for $3$: The sum of the digits of $425$ is $4 + 2 + 5 = 11$. Since $11$ is not divisible by $3$, $425$ is not divisible by $3$.
Check for $5$: The last digit of $425$ is $5$. According to the divisibility rule for $5$, any number ending in $0$ or $5$ is divisible by $5$.
$425 \div 5 = 85$
Step 4: Continuing the factorization of the new quotient.
Now, we factorize $85$.
Check for $5$: The last digit of $85$ is $5$.
$85 \div 5 = 17$
Step 5: Identifying the final prime factor.
The number $17$ is a prime number, meaning its only factors are $1$ and itself.
$17 \div 17 = 1$
Step 6: Expressing the number as a product of prime factors.
Combining all the prime factors obtained from the divisions above:
$3825 = 3 \times 3 \times 5 \times 5 \times 17$
Using exponential notation to group identical factors:
$3825 = 3^2 \times 5^2 \times 17^1$
Final Answer: The prime factorization of $3825$ is $3^2 \times 5^2 \times 17$.
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(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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