GSEB Class 6 Maths Solutions Chapter 3 Playing With Numbers Exercise 3.5

Question 1. Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4, must also be divisible by 8.
(g) All numbers which are divisible by 8, must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
Answer:
(a) False statement. For example, 6 is divisible by 3 but not by 9. A number being divided by 3 doesn't automatically mean it can be divided by 9.
(b) True statement. If a number is divisible by 9, it implies it is also divisible by 3 because 9 is a multiple of 3. If a larger number divides something, then its factors must also divide it.
(c) False statement. For example, 12 is divisible by both 3 and 6, but it is not divisible by 18. Divisibility by 3 and 6 does not always mean divisibility by 18 unless the factors are coprime.
(d) True statement. If a number is divisible by two coprime numbers (like 9 and 10, since their greatest common factor is 1), then it must also be divisible by their product (9 x 10 = 90). If numbers share no common factors, then their combined multiplication will always divide the original number.
(e) False statement. For example, 4 and 9 are coprime numbers, but neither 4 nor 9 are prime numbers. Co-prime simply means they share no common factors other than 1, not that one of them has to be a prime number.
(f) False statement. For example, 12 is divisible by 4 but not by 8. Being divisible by 4 doesn't guarantee divisibility by 8, as 8 is a larger multiple of 4.
(g) True statement. If a number is divisible by 8, it must also be divisible by 4, because 8 is a multiple of 4. If a number can be divided by a bigger number, it can also be divided by that bigger number's factors.
(h) True statement. This is a property of divisibility. If a number divides two distinct numbers exactly, then it will certainly divide their sum exactly. For instance, 2 divides 4 and 6, and it also divides their sum (4+6=10).
(i) False statement. For example, 5 divides the sum of 6 and 4 (which is 10), but 5 does not exactly divide 6 or 4 separately. Just because a number divides the total doesn't mean it divides each part.
In simple words: This question tests your knowledge of divisibility rules. You need to remember when one number dividing another automatically means it divides its factors or multiples, and when it doesn't. Sometimes, simply because two numbers are related (like factors or multiples), doesn't mean divisibility transfers.

Exam Tip: To verify true/false statements in number theory, always try to think of a simple counterexample. If you can find just one case where the statement doesn't hold, then it's false.

Question 2. Here are two different factor trees for 60. Write the missing numbers.
Answer:
For factor tree (a):
60 is broken down into 6 and 10.
6 is broken down into 2 and 3 (so, the missing number is 3).
10 is broken down into 5 and 2 (so, the missing number is 2).
So, the prime factorization of 60 in this tree is \( 2 \times 3 \times 5 \times 2 \).
For factor tree (b):
60 is broken down into 30 and 2.
30 is broken down into 10 and 3.
10 is broken down into 2 and 5 (so, the missing numbers are 2 and 5).
So, the prime factorization of 60 in this tree is \( 2 \times 5 \times 2 \times 3 \times 2 \).
In simple words: When you make a factor tree, you keep breaking numbers down into smaller factors until you only have prime numbers left. You just need to find the numbers that complete each branch of the tree.

Exam Tip: Remember that the product of the numbers on the lower branches must equal the number on the branch above them. This helps you find the missing numbers.

Question 3. Which factors are not included in the prime factorization of a composite number?
Answer: The factor 1 and the number itself are not included in the prime factorization of a composite number. Prime factorization focuses on expressing a composite number as a product of only its prime factors.
In simple words: When you break a number down to its prime factors, you don't list "1" or the original number itself because they aren't prime. You only show the actual prime numbers that multiply to make it.

Exam Tip: Prime factorization helps you understand the building blocks of numbers. Always ensure all factors in your final expression are indeed prime numbers.

Question 4. Write the greatest 4-digit number and express it in terms of its prime factors.
Answer: The greatest four-digit number is 9999.
We determine its prime factors through division:

The prime factorization of 9999 is \( 3 \times 3 \times 11 \times 101 \).
In simple words: The biggest number you can write with four digits is 9999. To break it down into its prime parts, you divide it by prime numbers until you can't divide it anymore. The prime numbers you used are its prime factors.

Exam Tip: When finding prime factors, always start with the smallest prime numbers (2, 3, 5, 7...) and work your way up. This ensures you find all prime factors systematically.

Question 5. Write the smallest 5-digit number and express it in the form of its prime factors.
Answer: The smallest 5-digit number is 10000.
We determine its prime factors through division:

The prime factorization of 10000 is \( 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 \).
In simple words: The smallest number you can write using five digits is 10000. To find its prime factors, you repeatedly divide it by the smallest possible prime numbers (like 2 and 5) until you get 1.

Exam Tip: For numbers ending in 0, you can always divide by 2 and 5. This is a quick way to begin your prime factorization process.

Question 6. Find all the prime factors of 1729 and arrange them in ascending order Now state the relation, if any, between two consecutive prime factors.
Answer: We determine the prime factors of 1729 through division:

The prime factorization of 1729 is \( 7 \times 13 \times 19 \).
Now, in ascending order, the prime factors of 1729 are: 7, 13, 19.
We check the differences between consecutive factors:
\( 19 - 13 = 6 \)
\( 13 - 7 = 6 \)
The relation is that the difference between any two consecutive prime factors is 6.
In simple words: First, you find all the prime numbers that multiply to make 1729. Then, you put these prime numbers in order from smallest to biggest. Finally, you look at the numbers next to each other and subtract to see how far apart they are. In this case, each pair is 6 apart.

Exam Tip: When testing for prime factors, remember the divisibility rules for small primes (2, 3, 5). For larger primes, you might need to try division by hand or calculator. Keep checking until the quotient becomes 1.

Question 7. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
Answer: Let us consider the following examples to verify the statement:

In each case, the product is divisible by both 2 and 3. When a number is divisible by both 2 and 3, it is also divisible by 6.
Thus, it is confirmed that the product of three consecutive numbers is always divisible by 6.
In simple words: We took a few sets of three numbers that come one after another. We multiplied them to get a product. Then we checked if each product could be divided by both 2 and 3. Since all products were divisible by both, it means they are also divisible by 6.

Exam Tip: To prove divisibility by 6, you need to show divisibility by both 2 and 3. For any three consecutive numbers, at least one will be even (divisible by 2) and exactly one will be a multiple of 3. This guarantees the product is divisible by 6.

Question 8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
Answer: Let us consider the following examples to verify the statement:

Since each of the numbers formed from the last two digits of the sum is divisible by 4, all the sums are also divisible by 4.
Thus, the sum of two consecutive odd numbers is always divisible by 4.
In simple words: We took several pairs of odd numbers that are right next to each other and added them up. For each sum, we checked if the last two digits of that sum could be divided by 4. Since they all could, it proves that the sum of any two odd numbers in a row will always be divisible by 4.

Exam Tip: The divisibility rule for 4 states that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. This is a very useful rule for quickly checking divisibility by 4.

Question 9. In which of the following expressions, prime factorization has been done?
(a) \( 24 = 2 \times 3 \times 4 \)
(b) \( 56 = 7 \times 2 \times 2 \times 2 \)
(c) \( 70 = 2 \times 5 \times 7 \)
(d) \( 54 = 2 \times 3 \times 9 \)
Answer:
(a) \( 24 = 2 \times 3 \times 4 \): This is not a prime factorization because 4 is not a prime number. For a true prime factorization, every factor must be prime.
(b) \( 56 = 7 \times 2 \times 2 \times 2 \): This is a prime factorization. All the factors (7 and 2) are prime numbers.
(c) \( 70 = 2 \times 5 \times 7 \): This is a prime factorization. All the factors (2, 5, and 7) are prime numbers.
(d) \( 54 = 2 \times 3 \times 9 \): This is not a prime factorization because 9 is not a prime number. For a true prime factorization, all numbers used to build the product must be prime.
In simple words: Prime factorization means breaking a number down into a product using only prime numbers (numbers that can only be divided by 1 and themselves, like 2, 3, 5, 7, etc.). Any expression that includes a non-prime number (like 4 or 9) in its factors is not a prime factorization.

Exam Tip: Always double-check that every number in your factorization is a prime number. If you find a composite number, break it down further until all factors are prime.

Question 10. Determine if 25110 is divisible by 45. Hint: 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9.
Answer: To determine if 25110 is divisible by 45, we can use the hint provided: 5 and 9 are coprime numbers. If 25110 is divisible by both 5 and 9, then it is also divisible by their product, 45.
First, let's check for divisibility by 5:
A number is divisible by 5 if its unit's digit is 0 or 5. The unit's digit of 25110 is 0. So, 25110 is divisible by 5.
Next, let's check for divisibility by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits of 25110 is \( 2 + 5 + 1 + 1 + 0 = 9 \). Since 9 is divisible by 9, 25110 is divisible by 9.
Since 25110 is divisible by both 5 and 9, and 5 and 9 are coprime, it means 25110 is divisible by \( 5 \times 9 = 45 \).
In simple words: To see if 25110 can be divided by 45, we first check if it can be divided by 5 and by 9 separately. Since it ends in 0, it divides by 5. Since its digits add up to 9, it divides by 9. Because both are true, it also divides by 45.

Exam Tip: Remember that if a number is divisible by two numbers that are coprime (share no common factors other than 1), then it is also divisible by their product. This rule is very helpful for complex divisibility tests.

Question 11. 18 is divisible by both 2 and 3. It is also divisible by 2 x 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 x 6 = 24? If not, give an example to justify your answer.
Answer: No, we cannot say that the number must also be divisible by 4 x 6 = 24.
The reason is that 4 and 6 are not coprime numbers; they share a common factor of 2 (i.e., their greatest common factor is 2, not 1). For the rule "if a number is divisible by two numbers, it is divisible by their product" to hold, the two numbers must be coprime.
For example, let's consider the number 36. 36 is divisible by 4 (because \( 36 \div 4 = 9 \)) and 36 is also divisible by 6 (because \( 36 \div 6 = 6 \)). However, 36 is not divisible by 24 (because \( 36 \div 24 \) leaves a remainder).
Similarly, the number 12 is divisible by 4 and 6, but it is not divisible by 24.
In simple words: Just because a number can be divided by 4 and by 6 doesn't mean it can also be divided by 24 (which is 4 times 6). This is because 4 and 6 share a common factor (2). If the numbers don't share any factors besides 1, then multiplying them works, but not otherwise.

Exam Tip: The rule for divisibility by a product only applies when the individual factors are coprime. Always check if the factors share any common factors before applying the product rule.

Question 12. I am the smallest number, having four different prime factors. Can you find me?
Answer: To find the smallest number with four different prime factors, we need to use the smallest prime numbers available.
The smallest four prime numbers are 2, 3, 5, and 7.
To find the smallest number that has these as its prime factors, we simply multiply them together.
Required number = \( 2 \times 3 \times 5 \times 7 \)
Required number = \( 6 \times 5 \times 7 \)
Required number = \( 30 \times 7 \)
Required number = 210
So, the smallest number having four different prime factors is 210.
In simple words: To make the smallest possible number that has four different prime numbers as its building blocks, you pick the first four smallest prime numbers (2, 3, 5, and 7) and multiply them all together. This gives you 210.

Exam Tip: To find the smallest number with a given number of distinct prime factors, always multiply the smallest prime numbers together up to the required count. For the largest such number, you would multiply the largest prime numbers.