100 points Hannah factored the expression 24xy + 15y. Her work is shown below. 1. GCF (24xy, 15y) = 3y

2. 24xy/3 = 8xy, 15y/3 = 5y

3. 3(24xy + 15y)

What are the errors in Hannah’s work? Check all that apply.

She divided by 3 instead of the GCF.

She found the wrong GCF.

She divided wrongly.

She didn’t complete step 4 and use the distributive property to be sure the expressions are equivalent.

She didn’t undistribute the GCF from the original expression.

1. She divided by 3 instead of the GCF.

4. She didn’t complete step 4 and use the distributive property to be sure the expressions are equivalent.

5. She didn’t undistribute the GCF from the original expression.

Since

8

x

2

y

2

,

16

x

3

y

4

,

12

x

5

y

3

contain both numbers and variables, there are two steps to find the GCF (HCF). Find GCF for the numeric part then find GCF for the variable part.

Steps to find the GCF for

8

x

2

y

2

,

16

x

3

y

4

,

12

x

5

y

3

:

1. Find the GCF for the numerical part

8

,

16

,

12

2. Find the GCF for the variable part

x

2

,

y

2

,

x

3

,

y

4

,

x

5

,

y

3

3. Multiply the values together

Find the common factors for the numerical part:

8

,

16

,

12

The factors for

8

are

−

8

,

−

4

,

−

2

,

−

1

,

1

,

2

,

4

,

8

.

Tap for more steps...

−

8

,

−

4

,

−

2

,

−

1

,

1

,

2

,

4

,

8

The factors for

16

are

−

16

,

−

8

,

−

4

,

−

2

,

−

1

,

1

,

2

,

4

,

8

,

16

.

Tap for more steps...

−

16

,

−

8

,

−

4

,

−

2

,

−

1

,

1

,

2

,

4

,

8

,

16

The factors for

12

are

−

12

,

−

6

,

−

4

,

−

3

,

−

2

,

−

1

,

1

,

2

,

3

,

4

,

6

,

12

.

Tap for more steps...

−

12

,

−

6

,

−

4

,

−

3

,

−

2

,

−

1

,

1

,

2

,

3

,

4

,

6

,

12

List all the factors for

8

,

16

,

12

to find the common factors.

8

:

−

8

,

−

4

,

−

2

,

−

1

,

1

,

2

,

4

,

8

16

:

−

16

,

−

8

,

−

4

,

−

2

,

−

1

,

1

,

2

,

4

,

8

,

16

12

:

−

12

,

−

6

,

−

4

,

−

3

,

−

2

,

−

1

,

1

,

2

,

3

,

4

,

6

,

12

The common factors for

8

,

16

,

12

are

−

4

,

−

2

,

−

1

,

1

,

2

,

4

.

−

4

,

−

2

,

−

1

,

1

,

2

,

4

The GCF for the numerical part is

4

.

GCF

Numerical

=

4

Next, find the common factors for the variable part:

x

2

,

y

2

,

x

3

,

y

4

,

x

5

,

y

3

The factors for

x

2

are

x

⋅

x

.

x

⋅

x

The factors for

y

2

are

y

⋅

y

.

y

⋅

y

The factors for

x

3

are

x

⋅

x

⋅

x

.

x

⋅

x

⋅

x

The factors for

y

4

are

y

⋅

y

⋅

y

⋅

y

.

y

⋅

y

⋅

y

⋅

y

The factors for

x

5

are

x

⋅

x

⋅

x

⋅

x

⋅

x

.

x

⋅

x

⋅

x

⋅

x

⋅

x

The factors for

y

3

are

y

⋅

y

⋅

y

.

y

⋅

y

⋅

y

List all the factors for

x

2

,

y

2

,

x

3

,

y

4

,

x

5

,

y

3

to find the common factors.

x

2

=

x

⋅

x

y

2

=

y

⋅

y

x

3

=

x

⋅

x

⋅

x

y

4

=

y

⋅

y

⋅

y

⋅

y

x

5

=

x

⋅

x

⋅

x

⋅

x

⋅

x

y

3

=

y

⋅

y

⋅

y

The common factors for the variables

x

2

,

y

2

,

x

3

,

y

4

,

x

5

,

y

3

are

x

⋅

x

⋅

y

⋅

y

.

x

⋅

x

⋅

y

⋅

y

The GCF for the variable part is

x

2

y

2

.

GCF

Variable

=

x

2

y

2

Multiply the GCF of the numerical part

4

and the GCF of the variable part

x

2

y

2

.

4

x

2

y

2

(n^3 - n)

= n(n-1)(n+1)

since it is divisible by 3 and 2 ,because it is a multiple 3 consecutive number

if, n-1 is odd then n+1 is also odd, then it is a multiple of 9

but if, n-1 is even then n+1 is also even, then it is a multiple of 4

so, largest multiple = max(9*2, 4*3) = max(18, 12) = 18