Given: 70 300 2.1 Write the above numbers in prime factors 2.2 Hence determine the LCM and HCF of 70 and 300
The Correct Answer and Explanation is:
2.1 Prime Factorization
Prime factorization of 70:
- Start by dividing 70 by the smallest prime number, which is 2:
70÷2=3570 \div 2 = 3570÷2=35
So, 70 = 2 × 35. - Next, divide 35 by the smallest prime number, which is 5:
35÷5=735 \div 5 = 735÷5=7.
So, 35 = 5 × 7. - 7 is a prime number, so we stop here.
Thus, the prime factorization of 70 is:
70=2×5×770 = 2 × 5 × 770=2×5×7.
Prime factorization of 300:
- Start by dividing 300 by 2:
300÷2=150300 \div 2 = 150300÷2=150.
So, 300 = 2 × 150. - Divide 150 by 2 again:
150÷2=75150 \div 2 = 75150÷2=75.
So, 150 = 2 × 75. - Divide 75 by 3 (since 75 is divisible by 3):
75÷3=2575 \div 3 = 2575÷3=25.
So, 75 = 3 × 25. - Now, divide 25 by 5:
25÷5=525 \div 5 = 525÷5=5.
So, 25 = 5 × 5. - 5 is prime, so we stop here.
Thus, the prime factorization of 300 is:
300=22×3×52300 = 2^2 × 3 × 5^2300=22×3×52.
2.2 Determining the LCM and HCF
HCF (Highest Common Factor)
To find the HCF, identify the common prime factors with the smallest powers:
- Prime factors of 70: 21×51×712^1 \times 5^1 \times 7^121×51×71.
- Prime factors of 300: 22×31×522^2 \times 3^1 \times 5^222×31×52.
The common prime factors are 2 and 5. The lowest powers of these factors are:
- For 2: 212^121.
- For 5: 515^151.
Thus, the HCF is:
HCF=21×51=2×5=10HCF = 2^1 \times 5^1 = 2 \times 5 = 10HCF=21×51=2×5=10.
LCM (Least Common Multiple)
To find the LCM, take all prime factors from both numbers with the highest powers:
- Prime factors of 70: 21×51×712^1 \times 5^1 \times 7^121×51×71.
- Prime factors of 300: 22×31×522^2 \times 3^1 \times 5^222×31×52.
The LCM is obtained by taking the highest powers of all prime factors:
- For 2: 222^222.
- For 3: 313^131.
- For 5: 525^252.
- For 7: 717^171.
Thus, the LCM is:
LCM=22×31×52×71=4×3×25×7=2100LCM = 2^2 \times 3^1 \times 5^2 \times 7^1 = 4 \times 3 \times 25 \times 7 = 2100LCM=22×31×52×71=4×3×25×7=2100.
Final Answer
- HCF of 70 and 300: 10.
- LCM of 70 and 300: 2100.
Explanation
The HCF is the largest number that divides both 70 and 300. It is found by considering the common factors between the two numbers and selecting the smallest powers of those factors. The LCM is the smallest number that both 70 and 300 can divide into, and it is found by taking all prime factors and using the largest powers of each factor.
