LCM of 26 and 91 by Prime Factorization and Division Methods
The concept of LCM of 26 and 91 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Finding the least common multiple of two numbers is especially useful in dealing with fractions, scheduling, and class 10 board exam questions.
Understanding LCM of 26 and 91
The LCM of 26 and 91 (Least Common Multiple) is the smallest number that is exactly divisible by both 26 and 91. This concept is widely used in fraction addition/subtraction, solving time-based problems in maths, and exam revision sessions. Knowing how to find the LCM using prime factorization and the division method is an important skill for students preparing for school and competitive exams.
Formula Used in LCM of 26 and 91
The standard formula is: \( \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} \)
For manual calculation, the LCM can be found by taking all the prime factors from both numbers, using the highest powers present.
Finding the LCM of 26 and 91 - Stepwise Methods
Method 1: Prime Factorization
1. Find prime factors of both numbers.
91 = 7 × 13
2. List all unique prime factors (use each common factor only once, with its highest power).
3. Multiply these prime factors:
Therefore, LCM(26, 91) = 182.
Method 2: Division (Ladder) Method
1. Write numbers 26 and 91 side by side.
2. Divide both numbers by the smallest prime number that divides at least one of them. For each step, bring the quotient below.
3. Multiply all the divisors used: 2 × 7 × 13 = 182
So, LCM(26, 91) = 182.
LCM and HCF Relationship for 26 and 91
The relationship between LCM and HCF (Highest Common Factor) is vital in exams. The formula is:
LCM × HCF = Product of the numbers
For 26 and 91: HCF = 13 (since 13 is the highest number that divides both).
Check: 182 × 13 = 2366
Also, 26 × 91 = 2366. The relation holds true!
Here’s a helpful table to understand the LCM of 26 and 91 more clearly:
LCM and HCF Table for 26 and 91
| Number | Prime Factors | HCF | LCM |
|---|---|---|---|
| 26 | 2 × 13 | 13 | 182 |
| 91 | 7 × 13 |
This table shows the factor breakdown and makes it easy to check the answer during exams.
Worked Example – Solving LCM of 26 and 91
1. Write both numbers: 26 and 91.
2. Find prime factors:
91 = 7 × 13
3. LCM must include each prime factor at the highest power:
4. Multiply all the selected prime factors:
Practice Problems
- Find the LCM of 26 and another number, like 39 or 65.
- If HCF(26, x) = 13 and LCM(26, x) = 182, what is x?
- List the common multiples of 26 and 91 up to 500.
- What is the LCM of 26, 91, and 13?
Common Mistakes to Avoid
- Confusing LCM with HCF – remember LCM is the smallest number divisible by both, HCF is the largest that divides both.
- Omitting common primes or repeating shared factors in multiplication (use each common factor only once with the highest power).
- Not checking the answer using the product-HCF-LCM relationship.
Real-World Applications
The concept of LCM of 26 and 91 appears in areas such as adding/subtracting unlike fractions, synchronizing events (e.g., two lights blinking every 26 and 91 seconds), and timetable management. Vedantu helps students see how these maths skills apply both in exams and in real life.
We explored the idea of LCM of 26 and 91, how to apply it with step-by-step solutions, check the result with the HCF-LCM relationship, and reviewed real-life uses. Practice more with Vedantu to build strong foundations in such topics and gain confidence for your board exams.
Related Learning Resources
- Prime Numbers: Useful for understanding how to break down numbers for LCM.
- LCM and HCF: Deep dive into their relationship, formulas, and differences.
- Factors of 91: Discover factorization details for better LCM calculation.
- Fundamental Theorem of Arithmetic: Learn why unique prime factorization matters for LCM.
- Multiples: Understand the idea of common multiples for LCM problems.
- LCM by Prime Factorization Method: View more solved examples stepwise.
- HCF by Long Division Method: Useful for co-practicing HCF and verifying LCM answers.
- Properties of HCF and LCM: For proofs and deeper mathematical understanding.
- Tables 2 to 20: Quick reference for multiplication during exam time.
- Factors of a Number: Foundation for any type of factor-based problem.
FAQs on How to Find the LCM of 26 and 91
1. What is the LCM of 26 and 91?
The LCM (Least Common Multiple) of 26 and 91 is 182. It is the smallest number that is exactly divisible by both 26 and 91.
2. How do you find the LCM of 26 and 91 by prime factorization?
To find the LCM of 26 and 91 by prime factorization, first break down both numbers into their prime factors:
26 = 2 × 13
91 = 7 × 13
Then, take all the unique prime factors with the highest power: 2, 7, and 13. Multiply them to get the LCM: 2 × 7 × 13 = 182.
3. What is the relationship between the LCM and HCF of 26 and 91?
The relationship between the LCM (Least Common Multiple) and HCF (Highest Common Factor) of two numbers is expressed by the formula:
LCM × HCF = Product of the two numbers.
For 26 and 91, since their product is 26 × 91 = 2366 and LCM is 182, the HCF can be found as 2366 ÷ 182 = 13.
4. Can I use the division method to find the LCM of 26 and 91?
Yes, the division method (also called the ladder method) is a practical way to find the LCM. You simultaneously divide 26 and 91 by their common or prime factors until no further division is possible. Multiply all the divisors used to get the LCM. For 26 and 91, the division method also results in 182 as the LCM.
5. Is the LCM of 26 and 91 useful for board exams?
Yes, understanding how to find the LCM of numbers like 26 and 91 is important for board exams, especially in topics related to fractions, ratios, time, and problem-solving. It helps students solve questions efficiently using methods like prime factorization or division method.
6. Why is 91 not a multiple of 26, even if 91 is larger?
Although 91 is larger than 26, it is not a multiple of 26 because 26 does not divide 91 evenly. A multiple of a number is obtained by multiplying it by whole numbers. Since 91 ÷ 26 does not yield a whole number, 91 is not a multiple of 26.
7. Why do some students confuse LCM with HCF in exam questions?
Students sometimes confuse LCM (Least Common Multiple) with HCF (Highest Common Factor) because both involve factors and multiples of numbers. However, LCM is about the smallest number divisible by given numbers, while HCF is the largest number that divides the given numbers. Clear conceptual understanding and practice help avoid this confusion.
8. Can the LCM of 26 and 91 be less than either number?
No, the LCM (Least Common Multiple) of two numbers cannot be less than either of the numbers because it is defined as the smallest number that both given numbers divide into evenly. Therefore, it must be equal to or greater than the largest of the two numbers. For 26 and 91, the LCM is 182, which is greater than both.
9. Why is prime factorization preferred over listing multiples for finding LCM?
Prime factorization is preferred because it is more efficient, especially for larger numbers. Listing multiples can become time-consuming and error-prone. Prime factorization breaks down numbers into their prime components, making it easy to identify all factors needed to calculate the LCM accurately and quickly.
10. How can mistakes happen in LCM division method steps?
Mistakes in the division method often occur due to incorrect division by non-prime numbers, missing a common prime factor, or stopping division too early. To avoid errors, divide by the smallest common prime factor at each step and continue until no further division is possible.
11. What are the factors of 26 and 91?
The factors of 26 are 1, 2, 13, and 26. The factors of 91 are 1, 7, 13, and 91. Identifying these helps understand their prime factors and calculate the LCM or HCF effectively.
12. How is the LCM of 26 and 91 verified using the product and HCF?
The LCM of 26 and 91 can be verified using the formula LCM × HCF = Product of the numbers. Since 26 × 91 = 2366, and the HCF is 13, LCM = 2366 ÷ 13 = 182. This confirms the accuracy of the calculated LCM.
