Answer :
To find the greatest common factor (GCF) of the expression
[tex]$$1035 x^5 y^2, \quad 315 x^6, \quad 180 x^5, \quad 45 x^4,$$[/tex]
we proceed as follows:
1. GCF of the Coefficients:
The coefficients are 1035, 315, 180, and 45. The greatest common divisor (GCD) of these numbers is the largest number that divides all of them. After evaluating the coefficients, we find that
[tex]$$\gcd(1035, 315, 180, 45) = 45.$$[/tex]
2. GCF of the Variable Factors:
- For the variable [tex]$x$[/tex], the exponents in the terms are 5, 6, 5, and 4. The GCF for the variable part is obtained by taking the smallest (minimum) exponent. Hence, the common factor is
[tex]$$x^4.$$[/tex]
- For the variable [tex]$y$[/tex], note that [tex]$y$[/tex] appears only in the first term as [tex]$y^2$[/tex]. Since [tex]$y$[/tex] is not present in every term, it is not part of the GCF.
3. Combine the Results:
The overall GCF of the expression is the product of the GCF of the coefficients and the GCF of the [tex]$x$[/tex]-terms:
[tex]$$45 \cdot x^4 = 45 x^4.$$[/tex]
Thus, the greatest common factor of the given expression is
[tex]$$\boxed{45 x^4},$$[/tex]
which corresponds to option A.
[tex]$$1035 x^5 y^2, \quad 315 x^6, \quad 180 x^5, \quad 45 x^4,$$[/tex]
we proceed as follows:
1. GCF of the Coefficients:
The coefficients are 1035, 315, 180, and 45. The greatest common divisor (GCD) of these numbers is the largest number that divides all of them. After evaluating the coefficients, we find that
[tex]$$\gcd(1035, 315, 180, 45) = 45.$$[/tex]
2. GCF of the Variable Factors:
- For the variable [tex]$x$[/tex], the exponents in the terms are 5, 6, 5, and 4. The GCF for the variable part is obtained by taking the smallest (minimum) exponent. Hence, the common factor is
[tex]$$x^4.$$[/tex]
- For the variable [tex]$y$[/tex], note that [tex]$y$[/tex] appears only in the first term as [tex]$y^2$[/tex]. Since [tex]$y$[/tex] is not present in every term, it is not part of the GCF.
3. Combine the Results:
The overall GCF of the expression is the product of the GCF of the coefficients and the GCF of the [tex]$x$[/tex]-terms:
[tex]$$45 \cdot x^4 = 45 x^4.$$[/tex]
Thus, the greatest common factor of the given expression is
[tex]$$\boxed{45 x^4},$$[/tex]
which corresponds to option A.
