Q1. |
Express each number as a product of its prime factors: |
Ans. |
(i) 140=2×2×5×7. So, the prime factorization of 140 is 22×5×7. (ii) 156=2×2×3×13. So, the prime factorization of 156 is 22×3×13. (iii) 3825=3×5×5×7×11. So, the prime factorization of 3825 is 3×52×7×11. (iv) 5005=5×7×11×13. So, the prime factorization of 5005 is 5×7×11×13. (v) 7429 is a prime number, so its prime factorization is itself: 7429=7429. |
Q2. | Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. |
Ans. | (i) 26 and 91: Prime factors of 26=2×13 Prime factors of 91=7×13 HCF = Highest common factor = 13 (since it's the only common prime factor) LCM = Least common multiple =2×7×13=182 Now, let's verify if LCM×HCF equals the product of the two numbers: LCM×HCF=182×13=2366 Product of 26 and 91=26×91=2366 So, LCM×HCF=Product of the two numbers. It’s verified. (ii) 510 and 92: Prime factors of 510=2×3×5×17 Prime factors of 92=2×2×23 HCF = Highest common factor = 2 (since it's the only common prime factor) LCM = Least common multiple = 2×2×3×5×17×23=23460 Now, let's verify if LCM×HCF equals the product of the two numbers: LCM×HCF=23460×2=46920 Product of 510 and 92=510×92=46920 So, LCM×HCF=Product of the two numbers. It’s verified. (iii) 336 and 54: Prime factors of 336=2×2×2×2×3×7=24×3×7 Prime factors of 54=2×3×3×3=2×33 HCF = Highest common factor = 2×3=6 LCM = Least common multiple = 24×33×7=3024 Now, let's verify if LCM×HCF equals the product of the two numbers: LCM×HCF=3024×6=18144 Product of 336 and 54=336×54=18144 So, LCM×HCF=Product of the two numbers. It’s verified. |
Q3. | Find the LCM and HCF of the following integers by applying the prime factorization method. |
Ans. | (i) 12, 15, and 21: Prime factors of 12=2×2×3=22×3 Prime factors of 15=3×5=3×5 Prime factors of 21=3×7=3×7 HCF = Highest common factor = Common prime factors with the lowest exponent = 3 LCM = Least common multiple = All prime factors with the highest exponent =22×3×5×7=420(ii) 17, 23, and 29: Prime factors of 17=17 Prime factors of 23=23 Prime factors of 29=29 HCF = Highest common factor = 1 (since there are no common prime factors) LCM = Least common multiple = Product of all the numbers =17×23×29=11389(iii) 8, 9, and 25: Prime factors of 8=2×2×2=23 Prime factors of 9=3×3=32 Prime factors of 25=5×5=22 HCF = Highest common factor = Common prime factors with the lowest exponent = 1 (since there are no common prime factors) LCM = Least common multiple = All prime factors with the highest exponent =23×32×52=3600 |
Q4. | Given that HCF (306, 657) = 9, find LCM (306, 657). |
Ans. | To find the LCM of 306 and 657 when their HCF is known, we can use the formula as given below: LCM (a, b)=(a×b)HCF (a, b) So, HCF(306,657)=9 (Given) Now, we can calculate the LCM as follows: LCM(306,657)=(306×657)9 LCM(306,657)=200,5629 LCM(306,657)=22,284 So, the LCM of 306 and 657 is 22,284. |
Q5. | Check whether 6n can end with the digit 0 for any natural number n. |
Ans. | Yes, 6n can end with the digit 0 for certain natural numbers n. In fact, any natural number n that is a multiple of 5 will result in 6n ending with the digit 0. Here's why: 6n=6×n When n is a multiple of 5, it can be expressed as n=5k, where k is a natural number. Now, substituting this value into 6n: 6n=6×(5k)=30k As we can see, 6n is equal to 30k, which ends with the digit 0. So, for any natural number n that is a multiple of 5, 6n will end with the digit 0. It’s verified. |
Q6. | Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers. |
Ans. | (i) \(7 \times 11 \times 13 + 13\): We can be simplified this expression as follows: 7×11×13+13=1001+13=1014 1014 has factors other than 1 and itself which are 2, 3, 6, 169, 338 and 507. Because it has more then two factors, It is a composite numbers. (ii) \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5\): We can be simplified this expression as follows: 7×6×5×4×3×2×1+5=5040+5=5045 5045 has factors other than 1 and itself which are 5 and 1009. Because it has more then two factors, It is a 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? |
Ans. | To find out when Sonia and Ravi will meet again at the starting point, we must calculate the LCM of their individual times to complete one round of the field. Given : Sonia takes 18 minutes to complete one round Ravi takes 12 minutes to complete one round The LCM of 18 and 12 will give us the time it takes for them to meet again at the starting point. So, firstly we find LCM. Prime factors of 18:2×3×3 (by prime factorization) Prime factors of 12:2×2×3 (by prime factorization) Now, calculate the LCM by taking the highest power of each prime factor: LCM(18,12)=22×32=4×9=36 Sonia and Ravi will meet again at the starting point after 36 minutes. |
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