Worksheet Solutions: Prime Time - 1

Multiple Choice Questions (MCQs)

Q1: Which of the following is a prime number?
(a) 49
(b) 51
(c) 53
(d) 55
Ans: (c) 53
Solution: 53 is a prime number as it has no divisors other than 1 and 53.

Q2: What is the first common multiple of 3 and 5?
(a) 10
(b) 12
(c) 15
(d) 20
Ans: (c) 15
Solution: The first common multiple of 3 and 5 is 15 because it is the smallest number that both 3 and 5 can divide into without a remainder.

Q3: How many prime numbers are there between 1 and 10?
(a) 2
(b) 3
(c) 4
(d) 5
Ans: (c) 4
Solution: The prime numbers between 1 and 10 are 2, 3, 5, and 7, making a total of 4.

Q4: Which pair of numbers is co-prime?
(a) 12 and 18
(b) 14 and 21
(c) 8 and 9
(d) 10 and 20
Ans: (c) 8 and 9
Solution: 8 and 9 are co-prime because they have no common factors other than 1.

Q5: The smallest number that is a multiple of both 3 and 4 is:
(a) 6
(b) 9
(c) 12
(d) 15
Ans: (c) 12
Solution: 12 is the smallest number that is a multiple of both 3 and 4 because it is the lowest common multiple of these two numbers.

Fill in the Blanks

Q1: The smallest prime number is _____.
Ans: 2
Solution: The smallest prime number is 2, which is also the only even prime number. All other even numbers are composite.

Q2: Numbers that have only two factors, 1 and the number itself, are called _____.
Ans: Prime numbers
Solution: A prime number is a number that can only be divided evenly by 1 and itself, such as 3, 5, 7, etc.

Q3: The common multiples of 3 and 5 within the first 100 numbers are _____, _____, and _____.
Ans: 15, 30, 45, 60, 75, 90
Solution: A common multiple of two numbers is a number that is a multiple of both. For 3 and 5, the common multiples include 15, 30, 45, 60, 75, 90

Q4: The Sieve of _____ is a method used to find all prime numbers up to a certain number.
Ans: Eratosthenes
Solution: The Sieve of Eratosthenes is an ancient algorithm used to find all primes up to a specified integer by progressively marking the multiples of each prime starting from 2.

Q5: Numbers that are neither prime nor composite are _____.
Ans: 1
Solution: The number 1 is unique because it only has one factor, itself, and is thus neither prime nor composite.

True or False

Q1: 9 is a prime number.
Ans: False
Solution: 9 is not a prime number because it can be divided by 1, 3, and 9.

Q2: The number 2 is the only even prime number.
Ans: True
Solution: 2 is the only even prime number because it can only be divided evenly by 1 and 2.

Q3: All multiples of 4 are also multiples of 2.
Ans: True
Solution: Every multiple of 4 is also a multiple of 2 because 4 is divisible by 2.

Q4: If a number is divisible by 8, it is also divisible by 4.
Ans: True
Solution: Since 8 is divisible by 4, any number divisible by 8 will also be divisible by 4.

Q5: The number 37 is a composite number.
Ans: False
Solution: 37 is a prime number as it only has two factors: 1 and 37.

Answer the following questions

Q1: List all the prime numbers between 10 and 20.
Ans: 11, 13, 17, 19
Solution: These numbers cannot be divided by any number other than 1 and themselves, making them prime.

Q2: Find the common factors of 24 and 36.
Ans: 1, 2, 3, 4, 6, 12
Solution: The common factors of 24 and 36 are the numbers that can divide both 24 and 36 without leaving a remainder.

Q3: What is the prime factorisation of 72?
Ans: 2 × 2 × 2 × 3 × 3
Solution: The prime factorisation of 72 involves breaking it down into its prime factors.

Q4: Identify two numbers between 1 and 50 that are co-prime.
Ans: 14 and 25
Solution: 14 and 25 are co-prime because they have no common factors other than 1.

Q5: What is the smallest multiple of 5 that is also a multiple of 3?
Ans: 15
Solution: 15 is the smallest number that is a multiple of both 5 and 3.

Q6.Is the first number divisible by the second? Use prime factorisation.

​a. 150 and 25

b. 84 and 12

Solution:

a. 150 and 25
Yes, 150 is divisible by 25.
Explanation: The prime factorisation of 150 is 2 × 3 × 5 × 5, and the prime factorisation of 25 is 5 × 5. Since 150 has all the factors of 25, it is divisible by 25.

b. 84 and 12
Yes, 84 is divisible by 12.
Explanation: The prime factorisation of 84 is 2 × 2 × 3 × 7, and the prime factorisation of 12 is 2 × 2 × 3. Since 84 has all the factors of 12, it is divisible by 12.

Q7: Find prime numbers, all less than 50, whose product is 2310.

Solution: The prime factorisation of 2310:
2310 = 2 × 3 × 5 × 7 × 11

All of the above numbers are prime.

Q8: What is the smallest number whose prime factorisation has:

a. Three different prime numbers?

b. Four different prime numbers?

Solution:

a. The smallest prime numbers are 2, 3, 5, and 7. To find the smallest number with these primes as factors, multiply them together:
2 x 3 × 5 = 30
So, the smallest number whose prime factorisation has three different prime numbers is 30.

b. The smallest four prime numbers are 2,3, 5 and 7. To find the smallest number with these primes as factors, multiply them together:
2 x 3 × 5 × 7 = 210
Thus, the smallest number whose prime factorisation has four different prime numbers is 210.

Q9: Who am I?
I am a number less than 50.
I am a multiple of 6.
The sum of my digits is 9.
I am greater than 30.

Solution: Multiples of 6 less than 50:
6, 12, 18, 24, 30, 36, 42, 48

Digit sum = 9 →
18 (1+8=9)
36 (3+6=9)

Number greater than 30 → 36

Q10. Which of the following numbers is the product of exactly three distinct prime numbers: 50, 84, 105, 280?

Solution:
Here,
50 = 2 × 5 × 5 (2 distinct primes)
84 = 2 × 2 × 3 × 7 (4 distinct primes)
105 = 3 × 5 × 7 (3 distinct primes)
280 = 2 × 2 × 2 × 5 × 7 (4 distinct primes)

Number 105 is the product of exactly three distinct prime numbers, i.e. 3 × 5 × 7.