GSEB Class 6 Maths Solutions Chapter 2 પૂર્ણ સંખ્યાઓ Exercise 2.3

Question 1. નીચેનામાંથી કોનો જવાબ શૂન્ય નથી?
(a) \( 1 + 0 \)
(b) \( 0 \times 0 \)
(c) \( \frac{0}{2} \)
(d) \( \frac{10-10}{2} \)
Answer: (a) \( 1 + 0 \)
(a) \( 1 + 0 = 1 \neq 0 \)
(b) \( 0 \times 0 = 0 \)
(c) \( \frac{0}{2} = 0 \times \frac{1}{2} = 0 \)
(d) \( \frac{10-10}{2} = \frac{0}{2} = 0 \)
Therefore, option (a) gives a result that is not zero.
In simple words: We need to find which calculation does not give zero. Adding zero to one gives one, which is not zero, making (a) the correct choice.

Exam Tip: Remember the basic rules for operations involving zero: adding zero to any number keeps the number, and multiplying any number by zero always results in zero. Division of zero by a non-zero number is zero.

Question 2. જો બે પૂર્ણ સંખ્યાઓનો ગુણાકાર શૂન્ય છે, તો શું આપણે કહી શકીએ છીએ કે આ સંખ્યાઓમાંથી એક કે શૂન્ય હોવી જોઈએ? ઉદાહરણ આપી ઉત્તર જણાવો.
Answer: Yes, if two whole numbers multiply to give zero, then one number or both numbers must be zero. We already know that multiplying any number by zero always gives zero.
For example:
1. \( 3 \times 0 = 0 \) and also \( 0 \times 3 = 0 \). The product is 0.
2. \( 7 \times 0 = 0 \) and also \( 0 \times 7 = 0 \). The product is 0.
3. \( 12 \times 0 = 0 \) and also \( 0 \times 12 = 0 \). The product is 0.
4. \( 0 \times 0 = 0 \) and also \( 0 \times 0 = 0 \). The product is 0.
In simple words: If you multiply two whole numbers and get zero, at least one of those numbers, or even both, must be zero. This is because anything multiplied by zero always equals zero.

Exam Tip: This concept is known as the "zero product property" in mathematics. It is fundamental for solving equations where a product equals zero.

Question 3. જો બે પૂર્ણ સંખ્યાઓનું ગુણનફળ 1 છે, તો શું આપણે કહી શકીએ છીએ કે આ સંખ્યાઓમાંથી એક કે બંને 1ના બરાબર હોવી જોઈએ? ઉદાહરણ આપી ઉત્તર જણાવો.
Answer: If the product of two whole numbers is 1, then both of those numbers must be 1.
For example:
Observe: \( 1 \times 1 = 1 \)
Also observe: \( 2 \times 1 = 2 \), \( 6 \times 1 = 6 \), \( 1 \times 8 = 8 \), \( 17 \times 1 = 17 \), and so on.
This means that for the product of two whole numbers to be 1, both numbers have to be 1.
In simple words: If you multiply two whole numbers and the answer is 1, then both numbers must be 1. There are no other whole number pairs that multiply to give 1.

Exam Tip: The number 1 is the multiplicative identity. When working with whole numbers, the only way to get a product of 1 is by multiplying 1 by itself.

Question 4. વિભાજનના ગુણધર્મનો ઉપયોગ કરી શોધોઃ
(a) \( 728 \times 101 \)
(b) \( 5437 \times 1001 \)
(c) \( 824 \times 25 \)
(d) \( 4275 \times 125 \)
(e) \( 504 \times 35 \)
Answer:
(a) \( 728 \times 101 = 728 \times (100 + 1) \)
\( = 728 \times 100 + 728 \times 1 \) (by distributive property)
\( = 72,800 + 728 \)
\( = 73,528 \)
(b) \( 5437 \times 1001 = 5437 \times (1000 + 1) \)
\( = 5437 \times 1000 + 5437 \times 1 \) (by distributive property)
\( = 54,37,000 + 5437 \)
\( = 54,42,437 \)
(c) \( 824 \times 25 = 824 \times (20 + 5) \)
\( = 824 \times 20 + 824 \times 5 \) (by distributive property)
\( = 16,480 + 4120 \)
\( = 20,600 \)
(d) \( 4275 \times 125 = 4275 \times (100 + 20 + 5) \)
\( = 4275 \times 100 + 4275 \times 20 + 4275 \times 5 \) (by distributive property)
\( = 4,27,500 + 85,500 + 21,375 \)
\( = 5,13,000 + 21,375 = 5,34,375 \)
(e) \( 504 \times 35 = 35 \times 504 \) (by commutative property of multiplication)
\( = 35 \times (500 + 4) \)
\( = 35 \times 500 + 35 \times 4 \) (by distributive property)
\( = 17,500 + 140 = 17,640 \)
In simple words: To multiply numbers more easily, we can break one of them into a sum of numbers, then multiply each part separately and add the results. This method is called the distributive property.

Exam Tip: The distributive property \( a \times (b + c) = (a \times b) + (a \times c) \) is a powerful tool for simplifying multiplication, especially with numbers close to powers of 10 like 101 or 99.

Question 5. નિગ્નલિખિત સ્વરૂપનું અધ્યયન કરોઃ
\( 1 \times 8 + 1 = 9 \)
\( 12 \times 8 + 2 = 98 \)
\( 123 \times 8 + 3 = 987 \)
\( 1234 \times 8 + 4 = 9876 \)
\( 12345 \times 8 + 5 = 98765 \)
આગળનાં બે પગથિયાં લખો. શું તમે કહી શકો છો કે સ્વરૂપ કઈ રીતે કાર્ય કરે છે?
Answer: From the above method, it is clear that the next two steps are:
\( 123456 \times 8 + 6 = 987654 \)
\( 1234567 \times 8 + 7 = 9876543 \)
Now, the method behind the pattern is as follows:
The number formed by consecutive digits can be expressed as a sum of numbers consisting of ones. For example, for 12345, the hint is: \( 12345 = 11111 + 1111 + 111 + 11 + 1 \).
So, let's observe the pattern with this understanding:
\( 1 \times 8 + 1 = 9 \)
\( 12 \times 8 + 2 = 98 = (11 + 1) \times 8 + 2 \)
\( 123 \times 8 + 3 = 987 = (111 + 11 + 1) \times 8 + 3 \)
\( 1234 \times 8 + 4 = 9876 = (1111 + 111 + 11 + 1) \times 8 + 4 \)
\( 12345 \times 8 + 5 = 98765 = (11111 + 1111 + 111 + 11 + 1) \times 8 + 5 \)
The next two steps are therefore:
\( 123456 \times 8 + 6 = 987654 = (111111 + 11111 + 1111 + 111 + 11 + 1) \times 8 + 6 \)
\( 1234567 \times 8 + 7 = 9876543 = (1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) \times 8 + 7 \)
In simple words: This pattern builds up by taking numbers like 1, 12, 123, and so on. We multiply by 8, then add the last digit of the first number. The result forms a sequence of 9, 98, 987, etc. The trick is how the starting numbers (like 12345) can be broken down into sums of numbers made only of ones, like \( 11111 + 1111 + 111 + 11 + 1 \).

Exam Tip: When analyzing patterns, always look for relationships between the input numbers and the output numbers. Sometimes, breaking down numbers into simpler components (like sums of ones) can reveal the underlying rule.