Q1. Here are two different factor trees for 60. Write the missing numbers.
Solutions:
Q2. Which factors are not included in the prime factorisation of a composite number?
Solutions:
1 and the number itself are not included in the prime factorisation of a composite number.
Q3. Write the greatest 4-digit number and express it in terms of its prime factors.
Solutions:
The greatest four digit number is 9999
Therefore 9999 = 3 × 3 × 11 × 101
Q4. Write the smallest 5-digit number and express it in the form of its prime factors.
Solutions:
The smallest five digit number = 10000
10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5
Q5. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.
Solutions:
1729 = 7 × 13 × 19
13 – 7 = 6
19 – 13 = 6
Hence, the difference between two consecutive prime factors is 6.
Q6. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
Solutions:
(i) 2 × 3 × 4 = 24 which is divisible by 6
(ii) 5 × 6 × 7 = 210 which is divisible by 6
Q7. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
Solutions:
(i) 5 + 3 = 8 which is divisible by 4
(ii) 7 + 9 = 16 which is divisible by 4
(iii) 13 + 15 = 28 which is divisible by 4
Q8. In which of the following expressions, prime factorisation has been done?
(a) 24 = 2 × 3 × 4
(b) 56 = 7 × 2 × 2 × 2
(c) 70 = 2 × 5 × 7
(d) 54 = 2 × 3 × 9
Solutions:
(a) 24 = 2 × 3 × 4
Since, 4 is composite. Hence, prime factorisation has not been done
(b) 56 = 7 × 2 × 2 × 2
Since, all the factors are prime. Hence, prime factorisation has been done
(c) 70 = 2 × 5 × 7
Since, all the factors are prime. Hence, prime factorisation has been done
(d) 54 = 2 × 3 × 9
Since, 9 is composite. Hence prime factorisation has not been done
Q9. 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.
Solutions:
No, since, 12 and 36 are both divisible by 4 and 6. But 12 and 36 are not divisible by 24
Q10. I am the smallest number, having four different prime factors. Can you find me?
Solutions:
Since, it is the smallest number. Therefore it will be the product of 4 smallest prime numbers
2 × 3 × 5 × 7 = 210
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