Show that any integer n > 12 can be written as a sum 4r + 5s for some nonnegative integers r, s. (this problem is sometimes called a postage stamp problem. it says that any postage greater than 11 cents can be formed using 4 cent and 5 cent stamps.)

Use induction for the prove

Step-by-step explanation:

Mathemathical induction is an useful method to prove things over natural numbers, you check for the first case, supose for the n and prove using your hypothesis for n+1

there says any integer bigger than 12 can be written as 4r+5s

so first number n can be is 13.

we can check n=13  =  4*2+5*1   r=2 and s=1 give 13.

Now we suppose n can be written as 4r+5s

and we can check if n+1=4r'+5s'  with  r' and s' integers.

we replace n as 4r+5s because that is our hypotesis

n+1=4r+5s+1

if we write that 1 as 5-4

4r+5s+1

4r+5s+5-4

then we can write

4(r-1)+5(s+1)   , we got n+1= 4 (r-1) +5(s+1)  where r-1 and s+1 are non negative integers. because r and s were no negative integers ( if r is not 0)

what if r=0?

if r is 0 , n is a multiple of 5   and n+1 can be written as 5s+1

first multiple of 5 we can write is 15 since n is bigger than 12 , then smaller s is 3.

for any n+1 we can write

n+1=5s+1=5 (s-3) +3*5+1=5(s-3)+4*4,   s-3 is 0 or bigger.

(check 3*5+1 is 16, the same as 4*4)

black/brown

step-by-step explanation:

well, you see here, jamal was born to 2 parents, his mother who he lives with and his father who left at a young age and who happens to be younger than him. he stole cookies because he has to steal food due to the fact that he spends his government poverty checks on pot and women.

solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.