GSEB Class 6 Maths Solutions Chapter 3 સંખ્યા સાથે Exercise 3.4

Question 1. સામાન્ય અવયવો શોધોઃ
(a) 20 અને 28
Answer:
20 = 1 × 20, 20 = 2 × 10, 20 = 4 × 5.
20ના બધા અવયવો : 1, 2, 4, 5, 10, 20
28 = 1 × 28, 28 = 2 × 14, 28 = 4 × 7
28ના બધા અવયવો: 1, 2, 4, 7, 14, 28 છે.
20 અને 28ના સામાન્ય અવયવો: 1, 2 અને 4
In simple words: First, list all the numbers that can divide 20 without leaving a remainder. Then, do the same for 28. Finally, pick the numbers that are in both lists.

Exam Tip: To find common factors, always list all factors for each number first, then identify the ones that appear in every list.

Question 1.
(b) 15 અને 25
Answer:
15 = 1 × 15, 15 = 3 × 5
15ના બધા અવયવો : 1, 3, 5, 15
25 = 1 × 25, 25 = 5 × 5.
25ના બધા અવયવો: 1, 5, 25
15 અને 25ના સામાન્ય અવયવો: 1 અને 5
In simple words: First, list all the numbers that can divide 15 without leaving a remainder. Then, do the same for 25. Finally, pick the numbers that are in both lists.

Exam Tip: Remember that 1 is always a common factor for any pair of numbers.

Question 1.
(c) 35 અને 50
Answer:
35 = 1 × 35, 35 = 5 × 7
35ના બધા અવયવો: 1, 5, 7, 35
50 = 1 x 50, 50 = 2 × 25, 50 = 5 × 10.
50ના બધા અવયવો : 1, 2, 5, 10, 25, 50
35 અને 50ના સામાન્ય અવયવો : 1 અને 5.
In simple words: First, list all the numbers that can divide 35 without leaving a remainder. Then, do the same for 50. Finally, pick the numbers that are in both lists.

Exam Tip: When finding factors, always start with 1 and the number itself, then check prime factors like 2, 3, 5, 7, and so on.

Question 1.
(d) 56 અને 120
Answer:
56 = 1 x 56, 56 = 2 × 28, 56 = 4 × 14, 56 = 7 × 8
56ના બધા અવયવો : 1, 2, 4, 7, 8, 14, 28, 56
120 = 1 x 120, 120 = 2 × 60, 120 = 3 × 40,
120 = 4 x 30, 120 = 5 × 24,
120 = 6 x 20, 120 = 8 × 15, 120 = 10 × 12
120ના બધા અવયવો : 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
56 અને 120ના સામાન્ય અવયવોઃ 1, 2, 4 અને 8
In simple words: To find common factors, list all the numbers that divide 56. Then, list all the numbers that divide 120. The numbers that show up in both lists are the common factors.

Exam Tip: Be systematic when listing factors for larger numbers to avoid missing any. Start from 1 and go up, checking divisibility.

Question 2. સામાન્ય અવયવો શોધો.
(a) 4, 8 અને 12
Answer:
4 = 1 × 4, 4 = 2 × 2.
4ના બધા અવયવો: 1, 2, 4
8 = 1 × 8, 8 = 2 × 4
8ના બધા અવયવો: 1, 2, 4, 8
12 = 1 x 12, 12 = 2 × 6, 12 = 3 x 4
12ના બધા અવયવો: 1, 2, 3, 4, 6, 12
4, 8 અને 12ના સામાન્ય અવયવો: 1, 2 અને 4
In simple words: Find all the numbers that can evenly divide 4, 8, and 12. The numbers that appear in the factor list for all three are the common factors.

Exam Tip: When finding common factors for three or more numbers, remember that a factor must be present in *all* numbers' factor lists.

Question 2.
(b) 5, 15 અને 25
Answer:
5 = 1 x 5
5ના બધા અવયવો: 1, 5
15 = 1 x 15, 15 = 3 × 5
15ના બધા અવયવો : 1, 3, 5, 15
25 = 1 x 25, 25 = 5 × 5
25ના બધા અવયવો: 1, 5, 25
5, 15 અને 25ના સામાન્ય અવયવો : 1 અને 5
In simple words: List all the numbers that divide 5, then 15, and then 25. The numbers that show up in all three lists are the common factors.

Exam Tip: Prime numbers like 5 only have two factors: 1 and themselves. This can simplify finding common factors.

Question 3. પ્રથમ ત્રણ સામાન્ય અવયવી શોધો:
(a) 6 અને 8.
Answer:
6ના અવયવીઃ 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
8ના અવયવી 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ... .
6 અને 8ના પ્રથમ ત્રણ સામાન્ય અવયવી : 24, 48 અને 72
In simple words: List the multiples of 6 (like 6, 12, 18...). Then list the multiples of 8 (like 8, 16, 24...). Find the first three numbers that are in both lists.

Exam Tip: To find common multiples, list out the multiples of each number until you find a few that appear in both lists. The Least Common Multiple (LCM) is the first one.

Question 3.
(b) 12 અને 18
Answer:
12ના અવયવીઃ 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...
18ના અવયવી : 18, 36, 54, 72, 90, 108, 126, ...
12 અને 18ના પ્રથમ ત્રણ સામાન્ય અવયવી 36, 72 અને 108
In simple words: Write down the multiples of 12 and 18. Then, find the first three numbers that appear in both lists of multiples. These are the common multiples.

Exam Tip: Starting with the LCM and then multiplying it by 1, 2, 3... is an easy way to find subsequent common multiples without listing everything again.

Question 4. 3 અને 4ના 100 કરતાં નાના સામાન્ય અવયવી લખો:
Answer:
3ના 100થી નાના અવયવી: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96 અને 99
4ના 100થી નાના અવયવીઃ 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92 અને 96
3 અને 4ના 100 કરતાં નાના હોય તેવા સામાન્ય અવયવી: 12, 24, 36, 48, 60, 72, 84 અને 96
In simple words: Find all the numbers that are multiples of both 3 and 4, and are smaller than 100. The smallest common multiple of 3 and 4 is 12, so you just need to list the multiples of 12 that are less than 100.

Exam Tip: If you need common multiples of two numbers, find their LCM first. All subsequent common multiples will be multiples of that LCM.

Question 5. નીચેની સંખ્યાઓમાંથી સહ-અવિભાજ્ય સંખ્યાઓ કઈ છે?
Answer:
સમજૂતી આપેલી બંને સંખ્યાઓના બધા અવયવો શોધો. જો બંને સંખ્યાના અવયવોમાં સામાન્ય અવયવ 1 હોય, તો આ બે સંખ્યાઓ સહ-અવિભાજ્ય સંખ્યાઓ કહેવાય.
In simple words: Co-prime numbers are pairs of numbers that have only one common factor, which is 1. If you list out all the factors for two numbers, and the only factor they share is 1, then they are co-prime.

Exam Tip: To check if numbers are co-prime, find their Greatest Common Divisor (GCD). If GCD is 1, they are co-prime.

Question 5.
(a) 18 અને 35
Answer:
18 = 1 x 18, 18 = 2 × 9, 18 = 3 × 6
18ના બધા અવયવો : 1, 2, 3, 6, 9 અને 18
35 = 1 × 35, 35 = 5 × 7
35ના બધા અવયવો: 1, 5, 7 અને 35
18 અને 35ના સામાન્ય અવયવ : 1
18 અને 35 સહ-અવિભાજ્ય સંખ્યાઓ છે.
In simple words: We find all the factors of 18 and 35. The only factor they both have is 1. Since 1 is their only common factor, 18 and 35 are co-prime numbers.

Exam Tip: Two numbers can be co-prime even if neither of them is a prime number, as long as they don't share any prime factors.

Question 5.
(b) 15 અને 37
Answer:
15 = 1 x 15, 15 = 3 × 5
15ના બધા અવયવો: 1, 3, 5 અને 15
37 = 1 x 37
37ના બધા અવયવો 1 અને 37
15 અને 37ના સામાન્ય અવયવ : 1
15 અને 37 સહ-અવિભાજ્ય સંખ્યાઓ છે.
In simple words: We find all the factors of 15 and 37. The only factor they share is 1. Since 1 is their only common factor, 15 and 37 are co-prime numbers.

Exam Tip: If one of the numbers is a prime number, it often simplifies checking for co-prime status, as its only factors are 1 and itself.

Question 5.
(c) 30 અને 415
Answer:
30 = 1 × 30, 30 = 2 × 15, 30 = 3 × 10, 30 = 5×6
30ના બધા અવયવો: 1, 2, 3, 5, 6, 10, 15 અને 30
415 = 1 x 415, 415 = 5 × 83
415ના બધા અવયવો: 1, 5, 83 અને 415
30 અને 415ના સામાન્ય અવયવો : 1 અને 5
30 અને 415 સહ-અવિભાજ્ય સંખ્યાઓ નથી.
In simple words: We find all the factors of 30 and 415. They both share the factors 1 and 5. Because they have more than just 1 as a common factor, 30 and 415 are not co-prime numbers.

Exam Tip: If two numbers end in 0 and 5 respectively, they will always have 5 as a common factor, meaning they cannot be co-prime.

Question 5.
(d) 17 અને 68
Answer:
17 = 1 x 17
17ના બધા અવયવો: 1 અને 17
68 = 1 x 68, 68 = 2 × 34, 68 = 4 × 17
68ના બધા અવયવો: 1, 2, 4, 17, 34 અને 68
17 અને 68ના સામાન્ય અવયવો : 1 અને 17
17 અને 68 સહ-અવિભાજ્ય સંખ્યાઓ નથી.
In simple words: We find all the factors of 17 and 68. They both share the factors 1 and 17. Since they have more than just 1 as a common factor, 17 and 68 are not co-prime numbers.

Exam Tip: If one number is a multiple of the other (e.g., 68 is a multiple of 17), then they will share the smaller number itself as a common factor, and thus won't be co-prime.

Question 5.
(e) 216 અને 25
Answer:
216 = 1 x 216, 216 = 2 × 108, 216 = 3 x 72, 216 = 4 × 54, 216 = 6 × 36, 216 = 8 × 27, 216 = 9 × 24, 216 = 12 x 18
216ના બધા અવયવો: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
25 = 1 x 25, 25 = 5 × 5
25ના બધા અવયવો: 1, 5, 25
216 અને 25ના સામાન્ય અવયવ : 1
216 અને 25 સહ-અવિભાજ્ય સંખ્યાઓ છે.
In simple words: We list all the factors for 216 and 25. The only factor they both have in common is 1. Therefore, 216 and 25 are considered co-prime numbers.

Exam Tip: To confirm co-prime status, it's efficient to check for common prime factors. If there are no shared prime factors, the numbers are co-prime.

Question 5.
(f) 81 અને 16.
Answer:
81 = 1 x 81, 81 = 3 x 27, 81 = 9 × 9,
81ના બધા અવયવો : 1, 3, 9, 27 અને 81
16 = 1 x 16, 16 = 2 × 8, 16 = 4 × 4
16ના બધા અવયવો : 1, 2, 4, 8 અને 16.
81 અને 16ના સામાન્ય અવયવ : 1
81 અને 16 સહ-અવિભાજ્ય સંખ્યાઓ છે.
In simple words: We list all the factors of 81 and 16. The only number they both can be divided by is 1. Because their only shared factor is 1, 81 and 16 are co-prime numbers.

Exam Tip: When checking for co-prime numbers, look for any shared prime factors. If none exist, the numbers are co-prime.

Question 6. એક સંખ્યા 5 અને 12 વડે વિભાજ્ય છે, તો તે સંખ્યા બીજી કઈ સંખ્યા વડે વિભાજ્ય છે?
Answer:
આપેલી સંખ્યાને 5 અને 12 બંને વડે નિઃશેષ ભાગાકાર કરી શકાય છે.
આપેલી સંખ્યાને 5 × 12 = 60 વડે પણ નિઃશેષ ભાગી શકાય. આમ, આ સંખ્યા 60 વડે પણ વિભાજ્ય છે.
નોંધઃ જો કોઈ સંખ્યાને બે સહ-અવિભાજ્ય સંખ્યાઓ વડે નિઃશેષ ભાગી શકાય, તો તે સંખ્યાને આ બે સહ-અવિભાજ્ય સંખ્યાઓના ગુણાકાર વડે પણ નિઃશેષ ભાગી શકાય.
In simple words: If a number can be divided perfectly by two different numbers, and those two numbers are co-prime (like 5 and 12), then the original number can also be divided perfectly by the product of those two numbers. Here, \( 5 \times 12 = 60 \), so the number must also be divisible by 60.

Exam Tip: If a number is divisible by two co-prime numbers, it is also divisible by their product. This is a fundamental property of divisibility rules.

Question 7. એક સંખ્યા 12 વડે વિભાજ્ય છે, તો તે સંખ્યા બીજી કઈ સંખ્યા વડે વિભાજ્ય છે?
Answer:
આપેલ સંખ્યાને 12 વડે નિઃશેષ ભાગી શકાય છે.
આ સંખ્યાને 12ના અવયવો વડે પણ નિઃશેષ ભાગી શકાય. એટલે કે, 12 = 1 × 12, 12 = 2 × 6, 12 = 3 × 4 આમ, 12ના અવયવો, 1, 2, 3, 4, 6 અને 12 છે.
આપેલી સંખ્યા 1, 2, 3, 4 અને 6 વડે વિભાજ્ય છે.
In simple words: If a number can be divided perfectly by 12, it means it can also be divided perfectly by all the factors of 12. The factors of 12 are 1, 2, 3, 4, and 6. So, the given number will also be divisible by these numbers.

Exam Tip: If a number is divisible by a composite number, it is also divisible by all the factors of that composite number. For instance, if divisible by 12, it's also divisible by 2, 3, 4, and 6.