Samacheer Kalvi Class 9 Maths Solutions Chapter 3 Algebra Exercise 3.9

Question 1. Find the GCD for the following:
(i) \( p^5, p^{11}, p^3 \)
Answer:
\( p^5 = p^5 \)
\( p^{11} = p^{11} \)
\( p^9 = p^9 \)
The Greatest Common Divisor (GCD) for these terms is \( p^5 \). This is because \( p^5 \) is the lowest power common to all given terms. Always look for the smallest exponent when finding the GCD of terms with the same base.
In simple words: To find the GCD of terms with the same letter and different powers, pick the letter with the smallest power.

🎯 Exam Tip: When finding the GCD of variables with exponents, always select the lowest power of the common variable present in all terms.

Question 1. Find the GCD for the following:
(ii) \( 4x^3, y^3, z^3 \)
Answer:
\( 4x^3 = 2 \times 2 \times x^3 \)
\( y^3 = y^3 \)
\( z^3 = z^3 \)
The GCD of \( 4x^3, y^3, \) and \( z^3 \) is 1. Since there are no common factors, other than 1, among the numbers (4) and the variables (x, y, z), the GCD is 1. This means the terms are relatively prime.
In simple words: If numbers and letters in different terms do not share any common factors, their GCD is 1.

🎯 Exam Tip: Remember that if terms have no common prime factors or common variables, their GCD is always 1, signifying they are coprime.

Question 1. Find the GCD for the following:
(iii) \( 9a^2b^2c^3, 15a^3b^2c^4 \)
Answer:
\( 9a^2b^2c^3 = 3 \times 3 \times a^2 \times b^2 \times c^3 \)
\( 15a^3b^2c^4 = 3 \times 5 \times a^3 \times b^2 \times c^4 \)
G.C.D \( = 3 \times a^2 \times b^2 \times c^3 \)
\( = 3a^2b^2c^3 \)
To find the GCD, we take the lowest power of each common prime factor and common variable. The common factors for 9 and 15 is 3, for \( a \) is \( a^2 \), for \( b \) is \( b^2 \), and for \( c \) is \( c^3 \). Combining these gives the final GCD.
In simple words: For each number and letter, find the biggest factor they both share, then multiply them together to get the GCD.

🎯 Exam Tip: Break down each coefficient into its prime factors and identify the lowest power of each common variable to correctly calculate the GCD.

Question 1. Find the GCD for the following:
(iv) \( 64x^8, 240x^6 \)
Answer:
Prime factorization of 64:
\( 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 \)
So, \( 64x^8 = 2^6 \times x^8 \)
Prime factorization of 240:
\( 240 = 2 \times 120 = 2 \times 2 \times 60 = 2 \times 2 \times 2 \times 30 = 2 \times 2 \times 2 \times 2 \times 15 = 2^4 \times 3 \times 5 \)
So, \( 240x^6 = 2^4 \times 3 \times 5 \times x^6 \)
G.C.D. \( = 2^4 \times x^6 \)
\( = 16x^6 \)
The common prime factor is 2, with the lowest power being \( 2^4 \). The common variable is \( x \), with the lowest power being \( x^6 \). Multiplying these together gives the GCD. This method is systematic for finding GCD of larger numbers.
In simple words: First, find the prime factors for the numbers, then find the smallest common power for both the numbers and the letters, and multiply them.

🎯 Exam Tip: Always use prime factorization for coefficients to ensure you capture all common factors, and then combine them with the lowest powers of common variables.

Question 1. Find the GCD for the following:
(v) \( ab^2c^3, a^2b^3c, a^3bc^2 \)
Answer:
\( ab^2c^3 = a \times b^2 \times c^3 \)
\( a^2b^3c = a^2 \times b^3 \times c \)
\( a^3bc^2 = a^3 \times b \times c^2 \)
G.C.D. \( = a \times b \times c \)
\( = abc \)
For each variable, we select the lowest exponent present across all terms. For 'a', the lowest power is \( a^1 \), for 'b' it is \( b^1 \), and for 'c' it is \( c^1 \). Multiplying these common factors gives the GCD. This is a quick way to find common factors among variable terms.
In simple words: For terms with different letters and powers, take the smallest power for each letter that is in all terms, and multiply them.

🎯 Exam Tip: When all terms share all variables, the GCD is simply the product of each variable raised to its lowest individual power across the terms.

Question 1. Find the GCD for the following:
(vi) \( 35x^5y^3z^4, 49x^2yz^3, 14xy^2z^2 \)
Answer:
\( 35x^5y^3z^4 = 5 \times 7 \times x^5 \times y^3 \times z^4 \)
\( 49x^2yz^3 = 7 \times 7 \times x^2 \times y \times z^3 \)
\( 14xy^2z^2 = 2 \times 7 \times x \times y^2 \times z^2 \)
G.C.D. \( = 7 \times x \times y \times z^2 \)
\( = 7xyz^2 \)
For the numerical coefficients (35, 49, 14), the greatest common factor is 7. For the variable 'x', the lowest power is \( x^1 \). For 'y', the lowest power is \( y^1 \). For 'z', the lowest power is \( z^2 \). Multiply these together to get the GCD. It's important to be careful with exponents for each variable.
In simple words: Find the biggest number that divides 35, 49, and 14. Then, for each letter (x, y, z), find the smallest power it has in all three parts. Multiply all these common parts together.

🎯 Exam Tip: Systematically find the GCD for coefficients and then for each variable separately by choosing the lowest exponent, and finally multiply all these common elements.

Question 1. Find the GCD for the following:
(vii) \( 25ab^3c, 100a^2bc, 125ab \)
Answer:
\( 25ab^3c = 5 \times 5 \times a \times b^3 \times c \)
\( 100a^2bc = 2 \times 2 \times 5 \times 5 \times a^2 \times b \times c \)
\( 125ab = 5 \times 5 \times 5 \times a \times b \)
G.C.D. \( = 5 \times 5 \times a \times b \)
\( = 25ab \)
The numerical coefficients (25, 100, 125) have a GCD of 25. For the variable 'a', the lowest power is \( a^1 \). For 'b', the lowest power is \( b^1 \). The variable 'c' is not present in all terms, so it is not part of the GCD. Therefore, the common product is \( 25ab \). It's crucial to only include variables present in *all* given terms.
In simple words: The largest number that divides 25, 100, and 125 is 25. The letter 'a' is in all parts, and its smallest power is 'a'. The letter 'b' is in all parts, and its smallest power is 'b'. The letter 'c' is not in the last part, so it is not included in the GCD.

🎯 Exam Tip: A variable is only included in the GCD if it appears in *every* single term; otherwise, it is skipped. Also, ensure the numerical GCD is found correctly.

Question 1. Find the GCD for the following:
(viii) \( 3abc, 5xyz, 7pqr \)
Answer:
\( 3abc = 3 \times a \times b \times c \)
\( 5xyz = 5 \times x \times y \times z \)
\( 7pqr = 7 \times p \times q \times r \)
G.C.D. \( = 1 \)
The coefficients (3, 5, 7) are all prime numbers and have no common factor other than 1. The variables (a, b, c), (x, y, z), and (p, q, r) are completely different sets and share no common letters. Therefore, the Greatest Common Divisor is 1. When terms are entirely unrelated, their GCD is always 1.
In simple words: Since there are no common numbers or letters shared by all three terms, their Greatest Common Divisor is simply 1.

🎯 Exam Tip: If the terms have no common numerical factors (other than 1) and no common variables, their GCD is 1, indicating they are mutually coprime.

Question 2. Find the GCD for the following:
(i) \( (2x + 5), (5x + 2) \)
Answer:
\( (2x + 5) = (2x + 5) \)
\( (5x + 2) = (5x + 2) \)
G.C.D. \( = 1 \)
These two binomials do not share any common factors. They are considered irreducible expressions in this context. Since there are no common terms or factors between \( (2x+5) \) and \( (5x+2) \), their GCD is 1. This means they are prime to each other.
In simple words: Since the two expressions cannot be broken down or simplified to share common parts, their greatest common factor is 1.

🎯 Exam Tip: For algebraic expressions, if no common factors can be factored out from both expressions, their GCD is 1.

Question 2. Find the GCD for the following:
(ii) \( a^{m+1}, a^{m+2}, a^{m+3} \)
Answer:
\( a^{m+1} = a^m \times a^1 \)
\( a^{m+2} = a^m \times a^2 \)
\( a^{m+3} = a^m \times a^3 \)
G.C.D.\( = a^{m+1} \)
When terms have the same base but different exponents, the GCD is the base raised to the smallest exponent. In this case, \( (m+1) \) is the smallest exponent among \( (m+1), (m+2), \) and \( (m+3) \). This property helps simplify finding GCDs with algebraic exponents.
In simple words: For terms with the same letter and power written like 'm+1', find the smallest power among them. That smallest power will be the GCD.

🎯 Exam Tip: For variables with exponents like \( a^{x} \), the GCD is always the term with the lowest exponent, as it is a factor of all higher powers.

Question 2. Find the GCD for the following:
(iii) \( 2a^2 + a, 4a^2 - 1 \)
Answer:
Factorize the first expression:
\( 2a^2 + a = a(2a + 1) \)
Factorize the second expression using the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \):
\( 4a^2 - 1 = (2a)^2 - 1^2 \)
\( = (2a + 1)(2a - 1) \)
G.C.D. \( = (2a + 1) \)
After factoring both expressions, we look for common factors. Both expressions share the factor \( (2a + 1) \). This is the only common part, so it is the GCD. Understanding algebraic identities is key here.
In simple words: Break down each expression into its simplest parts (factors). The part that is found in both broken-down expressions is the GCD.

🎯 Exam Tip: Always factorize algebraic expressions completely before attempting to find their GCD; look for common monomial factors and apply algebraic identities like the difference of squares.

Question 2. Find the GCD for the following:
(iv) \( 3a^2, 5b^3, 7c^4 \)
Answer:
\( 3a^2 = 3 \times a^2 \)
\( 5b^3 = 5 \times b^3 \)<
\( 7c^4 = 7 \times c^4 \)
G.C.D. \( = 1 \)
The coefficients (3, 5, 7) are all prime numbers and share no common factors other than 1. The variables \( (a^2), (b^3), \) and \( (c^4) \) are all different and do not share any common variable base. Because there are no common factors at all, the Greatest Common Divisor is 1. This means these terms are pairwise coprime.
In simple words: The numbers (3, 5, 7) and the letters (a, b, c) are all different and share no common factors. So, their GCD is 1.

🎯 Exam Tip: If numerical coefficients are coprime and all variables are distinct with no common bases, the GCD of the entire set of terms is 1.

Question 2. Find the GCD for the following:
(v) \( x^4 - 1, x^2 - 1 \)
Answer:
Factorize the first expression using the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \):
\( x^4 - 1 = (x^2)^2 - 1^2 \)
\( = (x^2 + 1)(x^2 - 1) \)
Further factorize \( (x^2 - 1) \):
\( = (x^2 + 1)(x + 1)(x - 1) \)
Factorize the second expression:
\( x^2 - 1 = (x + 1)(x - 1) \)
G.C.D. \( = (x + 1)(x - 1) \)
Both expressions share the factors \( (x+1) \) and \( (x-1) \). Therefore, their product, \( (x+1)(x-1) \) which is equal to \( x^2-1 \), is the GCD. Recognizing that one expression is a factor of the other simplifies the process.
In simple words: Break down both math problems into simpler parts using the difference of squares rule. The parts that are exactly the same in both answers make up the GCD.

🎯 Exam Tip: When one expression is a factor of the other (e.g., \( x^2-1 \) is a factor of \( x^4-1 \)), the smaller expression is usually the GCD if it divides the larger one completely.

Question 2. Find the GCD for the following:
(vi) \( a^3 - 9ax^2, (a - 3x)^2 \)
Answer:
Factorize the first expression:
\( a^3 - 9ax^2 \)
Take out the common factor 'a':
\( = a(a^2 - 9x^2) \)
Apply the difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \) to \( (a^2 - 9x^2) \):
\( = a[a^2 - (3x)^2] \)
\( = a(a + 3x)(a - 3x) \)
Factorize the second expression:
\( (a - 3x)^2 = (a - 3x)(a - 3x) \)
G.C.D. \( = (a - 3x) \)
After factoring both expressions, we find that the common factor between \( a(a + 3x)(a - 3x) \) and \( (a - 3x)(a - 3x) \) is \( (a - 3x) \). This is the only factor they share. Factoring completely helps identify all common elements.
In simple words: Factor out common parts from the first expression and use the difference of squares rule. The second expression is already factored. The common part between the two factored forms is the GCD.

🎯 Exam Tip: Always factor out any common monomials first, then apply algebraic identities for further factorization to fully reveal all common factors between expressions.