30 points prove that: a) 2^13–2^10–2^9 is divisible by 13 b)27^4–9^5+3^9 is divisible by 25 c) 16^4–2^13–4^5 is divisible by 11

a) done in the previous question.

See below for b and c.

Step-by-step explanation:

(b)  27^4 - 9^5 + 3^9.  

Change these terms to base 3:

=  3^(4*3) - 3^(2*5) + 3^9

= 3^12 - 3^10 + 3^9

= 3^9( 3^3 - 3 + 1))

= 25 * 3^9   so its divisible by 25.

c. 16^4–2^13–4^5    

 Change to base 2:

= 2^(4*4) - 2^13 - 2^(2*5)

= 2^16 - 2^13 - 2^10

= 2^10 (2^6 - 2^3 - 1)

= 2^10 ( 64 - 8 - 1)

= 55 * 2^10

= 11*5 *2^10  which is divisible by 11.

system of linear equation

step-by-step explanation:

as given in the data

a= 20 , b= -12

let us suppose that it is a straight line equation

now we know the formula for the system of linear equation

b= ma + c )

here m is slope of a straight line which is equal to formula

m=   (ii)

and here c is a constant

now putting the value of a and b in equation (ii)

m=

m=

putting the value of a,b and m in equation (i) gives us

-12 = + c

-12=-12 + c

adding 12 to both sides

-12 +12 = -12 +12 +c

0 =c

or c=0

as the value of the constant is zero which gives us that it is a linear equation and this order pair satisfies the system of linear equations.