Real Numbers (HCF LCM Euclid) Solved Examples (10 Mathematics)
Euclidean algorithm and HCF-LCM problems form the foundation of number theory in Class 10. These solved examples show how to find HCF using Euclid's algori
TL;DR: Euclidean algorithm and HCF-LCM problems form the foundation of number theory in Class 10. These solved examples show how to find HCF using Euclid's a…
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Euclidean algorithm and HCF-LCM problems form the foundation of number theory in Class 10. These solved examples show how to find HCF using Euclid's algori
Real Numbers (HCF LCM Euclid) — Solved Numerical Examples (Step by Step)
Example 1: Find the HCF of 378 and 294 using Euclid's division algorithm.
Solution: Apply Euclid's division algorithm: divide the larger number by the smaller.
378 = 294 x 1 + 84
294 = 84 x 3 + 42
84 = 42 x 2 + 0
When remainder becomes 0, the last divisor is the HCF.
Example 2: Find HCF and LCM of 96 and 404. Verify that HCF x LCM = a x b.
Solution: Step 1: Find HCF using Euclid's algorithm.
404 = 96 x 4 + 20
96 = 20 x 4 + 16
20 = 16 x 1 + 4
16 = 4 x 4 + 0
HCF = 4
Step 2: Use formula LCM = (a x b) / HCF
LCM = (96 x 404) / 4 = 38784 / 4 = 9696
Step 3: Verify HCF x LCM = 4 x 9696 = 38784
Also a x b = 96 x 404 = 38784
Verified: HCF x LCM = a x b
Example 3: Three bells ring at intervals of 24, 36, and 48 minutes. If they ring together at 9:00 AM, when will they ring together again?
Solution: The bells will ring together at the LCM of 24, 36, and 48.
Prime factorization:
24 = 2^3 x 3
36 = 2^2 x 3^2
48 = 2^4 x 3
LCM = 2^4 x 3^2 = 16 x 9 = 144 minutes
Convert to hours and minutes: 144 minutes = 2 hours 24 minutes
Time = 9:00 AM + 2 hours 24 minutes = 11:24 AM
Example 4: Show that 7 is a factor of 343 and find all divisors of 343.
Solution: Step 1: Check if 7 is a factor.
343 / 7 = 49, so 7 is indeed a factor.
Step 2: Find prime factorization.
343 = 7 x 49 = 7 x 7 x 7 = 7^3
Step 3: Find all divisors.
Divisors of 7^3 are: 7^0, 7^1, 7^2, 7^3
Which are: 1, 7, 49, 343
Example 5: Find the HCF of 1260 and 1728 using prime factorization.
Solution: Step 1: Prime factorize 1260.
1260 = 2^2 x 3^2 x 5 x 7
Step 2: Prime factorize 1728.
1728 = 2^6 x 3^3
Step 3: HCF is product of lowest powers of common prime factors.
Common primes: 2 and 3
HCF = 2^2 x 3^2 = 4 x 9 = 36
Example 6: Two ropes are 30 m and 45 m long. They need to be cut into equal pieces without waste. What is the maximum length of each piece?
Solution: Maximum length of each piece = HCF of 30 and 45
45 = 30 x 1 + 15
30 = 15 x 2 + 0
HCF = 15 m
Verification:
30 m rope will have 30/15 = 2 pieces
45 m rope will have 45/15 = 3 pieces
Total = 5 pieces of 15 m each
Example 7: If HCF(a, b) = 12 and LCM(a, b) = 720, find a and b if a = 48.
Solution: Use the relationship: HCF x LCM = a x b
12 x 720 = 48 x b
8640 = 48 x b
b = 8640 / 48 = 180
Verify: HCF(48, 180)
180 = 48 x 3 + 36
48 = 36 x 1 + 12
36 = 12 x 3 + 0
HCF = 12 ✓
LCM = (48 x 180) / 12 = 8640 / 12 = 720 ✓
Tips
- Always verify your HCF by checking that it divides both numbers exactly.
- Remember HCF x LCM = a x b; use this to check your work.
- When finding HCF of three or more numbers, find HCF of first two, then HCF of result with the third.
- For LCM by prime factorization, take the HIGHEST power of each prime factor that appears.
Frequently Asked Questions
Why does Euclid's algorithm work?
Euclid's algorithm works because the HCF of two numbers equals the HCF of the smaller number and the remainder. This is based on the property that any common divisor of a and b also divides their difference.
Can HCF ever be larger than LCM?
No, HCF is always less than or equal to LCM. In fact, HCF = LCM only when the two numbers are equal.
More Mathematics Solved Examples
- Pair of Linear Equations in Two Variables
- Circles
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- Linear Equations in Two Variables
- Heron's Formula
- Integrals (Indefinite and Definite)
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