Write the greatest common factor in each of the following terms.
(i) -18a², 108a
(ii) 3x²y, 18xy², -6xy
(iii) 2xy, -y², 2x²y
(iv) l²m²n, lm²n², l²mn²
(v) 21pqr, -7p²q²r², 49p²qr
(vi) qrxy, pryz, rxyz
(vii) 3x³y²z, -6xy³z², 12x²yz³
(viii) 63p²a²r²s, - 9pq²r²s², 15p²qr²s², - 60p²a²rs²
(ix) 13x²y, 169xy
(x) 11x², 12y²
Solution:
'Factor' is a term used to express a number as a product of any two numbers. Factorization is a method of finding factors for any mathematical object, be it a number, a polynomial or any algebraic expression.
The Greatest Common Factor(GCF) of two or more non-zero integers, x & y, is the greatest positive integer m, which divides both, x & y.
(i) -18a², 108a
Given terms are: -18a², 108a
Factors of -18a² = (-1) × 2 × 3 × 3 × a × a
Factors of 108a = 2 × 2 × 3 × 3 × 3 × a
∴ GCF = 2 × 3 × 3 × a = 18a
(ii) 3x²y, 18xy², -6xy
Given terms are 3x²y, 18xy², -6xy
Factors of 3x²y = 3 × x × x × y
Factors of 18xy² = 2 × 3 × 3 × x × y × y
Factors of - 6xy = (-1) × 2 × 3 × x × y
∴ GCF = 3xy
(iii) 2xy, -y², 2x²y
Given terms are 2xy, -y², 2x²y
Factors of 2xy = 2 × x × y
Factors of -y² = (-1) × y × y
Factors of 2x²y = 2 × x × x × y
∴ GCF = y
(iv) l²m²n, lm²n², l²mn²
Given terms are l²m²n, lm²n², l²mn²
Factors of l²m²n = l × l × m × m × n
Factors of lm²n² = l × m × m × n × n
Factors of l²mn² = l × l × m × n × n
∴ GCF = lmn
(v) 21pqr, -7p²q²r², 49p²qr
Given terms are 21pqr, -7p²q²r², 49p²qr
Factors of 21pqr = 3 × 7 × p × q × r
Factors of -7p²q²r² = (-1) × 7 × p × p × q × q × r × r
Factors of 49p²qr = 7 × 7 × p × p × q × r
∴ GCF = 7pqr
(vi) qrxy, pryz, rxyz
Given terms are qrxy, pryz, rxyz
Factors of qrxy = q × r × x × y
Factors of pryz = p × r × y × z
Factors of rxyz = r × x × y × z
∴ GCF = ry
(vii) 3x³y²z, -6xy³z², 12x²yz³
Given terms are 3x³y²z, -6xy³z², 12x²yz³
Factors of 3x³y²z = 3 × x × x × x × y × y × z
Factors of -6xy³z² = (-1) × 2 × 3 × x × y × y × y × z × z
Factors of 12x²yz³ = 2 × 2 × 3 × x × x × y × z × z × z
∴ GCF = 3xyz
(viii) 63p²a²r²s, - 9pq²r²s², 15p²qr²s², - 60p²a²rs²
Given terms are 63p²a²r²s, - 9pq²r²s², 15p²qr²s², - 60p²a²rs²
Factors of 63p²a²r²s = 3 × 3 × 7 × p × p × a × a × r × r × s
Factors of - 9pq²r²s² = (-1) × 3 × 3 × p × q × q × r × r × s × s
Factors of 15p²qr²s² = 3 × 5 × p × p × q × r × r × s × s
Factors of - 60p²a²rs² = (-1) × 2 × 2 × 3 × 5 × p × p × a × a × r × s × s
∴ GCF = 3prs
(ix) 13x²y, 169xy
Given terms are 13x²y, 169xy
Factors of 13x²y = 13 × x × x × y
Factors of 169xy = 13 × 13 × x × y
∴ GCF = 13xy
(x) 11x², 12y²
Factors of 11x² = 11× x × x
Factors of 12y² = 2 × 2 × 3 × y × y
Factors of 12y² =
∴ GCF = 1
✦ Try This: Write the greatest common factor in each of the following terms.
(i) a³b²c⁴, 12a²b⁴c³,36a²b²c²
(ii) 144p⁴a²r²s³, - 36p⁴q²r²s⁴, 72p³q³r³s³, - 60p²a²rs²
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 9
NCERT Exemplar Class 8 Maths Chapter 7 Problem 87
Write the greatest common factor in each of the following terms. (i) -18a², 108a, (ii) 3x²y, 18xy², -6xy, (iii) 2xy, -y², 2x²y, (iv) l²m²n, lm²n², l²mn², (v) 21pqr, -7p²q²r², 49p²qr, (vi) qrxy, pryz, rxyz, (vii) 3x³y²z, -6xy³z², 12x²yz³, (viii) 63p²a²r²s, - 9pq²r²s², 15p²qr²s², - 60p²a²rs², (ix) 13x²y, 169xy, (x) 11x², 12y²
Summary:
The greatest common factor in each of the following terms: (i) -18a², 108a, (ii) 3x²y, 18xy², -6xy, (iii) 2xy, -y², 2x²y, (iv) l²m²n, lm²n², l²mn², (v) 21pqr, -7p²q²r², 49p²qr, (vi) qrxy, pryz, rxyz, (vii) 3x³y²z, -6xy³z², 12x²yz³, (viii) 63p²a²r²s, - 9pq²r²s², 15p²qr²s², - 60p²a²rs², (ix) 13x²y, 169xy, (x) 11x², 12y² are 18a, 3xy, y, lmn, 7pqr, ry, 3xyz, 3prs, 13xy and 1 respectively
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