GSEB Class 6 Maths Solutions Chapter 3 Playing With Numbers Exercise 3.2

Question 1. What is the sum of any two:
(a) odd numbers?
(b) even numbers?
Answer:
(a) The sum of any two odd numbers is always an even number.
(b) The sum of any two even numbers is always an even number.
In simple words: When you add two odd numbers, you always get an even number. The same thing happens when you add two even numbers; the result is always even.

Exam Tip: Remember that adding two numbers with the same parity (both odd or both even) always results in an even number. This is a fundamental property of addition with integers.

Question 2. Stare whether the following statements are True or False:
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(c) The product of three odd numbers is odd.
(d) If an even number is divided by 2 the quotient is always odd.
(e) All prime numbers are odd.
(f) Prime numbers do not have any factors.
(g) Sum of two prime numbers is always even.
(h) 2 is the only even prime number
(i) All even numbers are composite numbers.
(j) The product of two even numbers is always even.
Answer:
(a) False
(b) True
(c) True
(d) False
(e) False
(f) False
(g) False (For instance, \( 2 + 3 = 5 \), which is an odd number.)
(h) True
(i) False (The number 2 is an even number but it is also a prime number.)
(j) True
In simple words: Review each statement about numbers. Check if it is really true or false by thinking of examples. Sometimes, the rule might not work for all numbers, making the statement false.

Exam Tip: For True/False questions about number properties, always test with a small example. For false statements, giving a counter-example (like \( 2 + 3 = 5 \) for g) can help clarify your reasoning.

Question 3. The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.
Answer: We know that the prime numbers up to 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 and 97.
Out of these prime numbers, except 13 and 31, the required pairs of prime numbers that have the same digits are:
(17 and 71); (37 and 73) and (79 and 97).
In simple words: Look for prime numbers below 100 that use the exact same digits, just in a different order. For example, 17 and 71 both use the digits 1 and 7.

Exam Tip: To find prime numbers efficiently, you can use the Sieve of Eratosthenes. When looking for pairs, carefully check the digits of each prime number found.

Question 4. Write down separately the prime and composite numbers less than 20.
Answer: The numbers below 20 are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19.
Since 1 is neither a prime nor a composite number, from these numbers, the prime numbers are: 2, 3, 5, 7, 11, 13, 17 and 19.
All remaining numbers are composite numbers.
The composite numbers below 20 are: 4, 6, 8, 9, 10, 12, 14, 15, 16 and 18.
In simple words: Write all numbers from 1 to 19. Then, sort them into two groups: numbers that are only divisible by 1 and themselves (primes), and numbers that have more than two factors (composites). Remember that 1 doesn't belong to either group.

Exam Tip: A prime number has exactly two factors (1 and itself). A composite number has more than two factors. The number 1 is unique and is neither prime nor composite.

Question 5. What is the greatest prime number between 1 and 10?
Answer: The prime numbers between 1 and 10 are 2, 3, 5, and 7.
Therefore, the greatest prime number between 1 and 10 is 7.
In simple words: Find all the special numbers between 1 and 10 that only divide by themselves and 1. Then, pick the biggest one.

Exam Tip: To find prime numbers, list integers and eliminate multiples of smaller primes (e.g., eliminate all even numbers except 2, all multiples of 3 except 3, etc.).

Question 6. Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18
Answer:
(a) \( 44 = 13 + 31 \) or \( 44 = 3 + 41 \) or \( 44 = 7 + 37 \)
(b) \( 36 = 5 + 31 \) or \( 36 = 23 + 13 \) or \( 36 = 17 + 19 \)
(c) \( 24 = 11 + 13 \) or \( 24 = 5 + 19 \) or \( 24 = 7 + 17 \)
(d) \( 18 = 7 + 11 \) or \( 18 = 5 + 13 \)
In simple words: For each number, find two prime numbers that are not even (except for 2, which is not used here as we need two odd primes) and add them together to get the given total.

Exam Tip: Keep a list of small prime numbers handy. Start by trying to subtract a small odd prime from the target number and check if the remainder is also an odd prime.

Question 7. Give three pairs of prime numbers whose difference is 2.
Answer: Two prime numbers whose difference is 2 are called twin primes.
Three pairs of prime numbers whose difference is 2 are:
(i) 3 and 5
(ii) 5 and 7
(iii) 11 and 13
In simple words: Find pairs of prime numbers that are very close, so close that if you subtract one from the other, you get exactly two.

Exam Tip: Twin primes are consecutive odd numbers that are both prime. You can look for them by checking odd numbers and their immediate odd successors.

Question 8. Which of the following numbers are prime?
(a) 23
(b) 51
(c) 37
(d) 26
Answer:
(a) \( 23 = 23 \times 1 \). The factors of 23 are 1 and 23 itself. Therefore, 23 is a prime number.
(b) \( 51 = 1 \times 51 = 3 \times 17 \). The factors of 51 are 1, 3, 17, and 51. Therefore, 51 is not a prime number.
(c) \( 37 = 1 \times 37 \). The factors of 37 are 1 and 37. Therefore, 37 is a prime number.
(d) \( 26 = 1 \times 26 = 2 \times 13 \). The factors of 26 are 1, 2, 13, and 26. Therefore, 26 is not a prime number.
In simple words: To find out if a number is prime, try dividing it by small numbers. If it only divides by 1 and itself, it's prime. If it divides by other numbers too, it's not prime.

Exam Tip: To check if a number is prime, only test for divisibility by prime numbers up to the square root of that number. If it's not divisible by any of these, it's prime.

Question 9. Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Answer: The required seven consecutive composite numbers less than 100 are:
90, 91, 92, 93, 94, 95, 96.
In simple words: Find a group of seven numbers in a row, all smaller than 100. Every single number in this group must be a composite number, meaning none of them can be prime.

Exam Tip: A composite number has more than two factors. To find consecutive composite numbers, look for gaps between prime numbers. The larger the numbers, the more likely you are to find longer sequences of composite numbers.

Question 10. Express each of the following numbers as the sum of three odd primes:
(a) 21
(b) 31
(c) 53
(d) 61
Answer:
(a) For 21, we have: \( 21 = 3 + 5 + 13 \)
(b) For 31, we have: \( 31 = 3 + 5 + 23 \)
(c) For 53, we have: \( 53 = 13 + 17 + 23 \)
(d) For 61, we have: \( 61 = 7 + 13 + 41 \)
In simple words: For each number, pick three odd prime numbers that, when added together, give you that exact total.

Exam Tip: This type of problem relies on Goldbach's weak conjecture (for odd numbers). Start with the smallest odd primes (3, 5, 7, etc.) and experiment with combinations. It often helps to keep a list of odd primes available.

Question 11. Write five pairs of prime numbers less than 20 whose sum is divisible by 5. Hint: 3 + 7 = 10
Answer: The prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17 and 19.
We have:
\( 2 + 3 = 5 \) and 5 is divisible by 5.
\( 3 + 7 = 10 \) and 10 is divisible by 5.
\( 2 + 13 = 15 \) and 15 is divisible by 5.
\( 7 + 13 = 20 \) and 20 is divisible by 5.
\( 3 + 17 = 20 \) and 20 is divisible by 5.
The required prime number pairs are:
(i) 2 and 3
(ii) 3 and 7
(iii) 2 and 13
(iv) 7 and 13
(v) 3 and 17
In simple words: Look at all the prime numbers smaller than 20. Find pairs of these numbers that, when added up, give a total that can be divided evenly by 5.

Exam Tip: To check for divisibility by 5, simply look at the last digit of the sum. If the last digit is 0 or 5, the number is divisible by 5.

Question 12. Fill in the blanks.
(a) A number which has only two factors is called a ..........................
(b) A number which has more than two factors is called a ..........................
(c) 1 is neither .......................... nor ..........................
(d) The smallest prime number is ..........................
(e) The smallest composite number is ..........................
(f) The smallest even number is ..........................
Answer:
(a) prime number
(b) composite number
(c) prime, composite
(d) 2
(e) 4
(f) 2
In simple words: Fill in the missing words for each sentence using the correct math terms about different types of numbers.

Exam Tip: Understand the definitions of prime, composite, even, and odd numbers. Remember the special case of the number 1.