hey um this isn't an answer but do you know the answer to this
Step-by-step explanation:
Arnold is planning on painting a mural. Before painting the mural, Arnold painted the picture on a paper that is 5 inches wide and 11 inches tall. Arnold is using a scale in which 1/2 inch represents 1 foot to paint the mural.
How tall will the actual mural be?
How wide will the actual mural be?
The four answers to this problem are: x = 19, 30, 47, 58 (mod 77)
Step-by-step explanation:
Note that 77 = (11)(7), so we can rewrite the equation as it follows
1)
And
2)
We will work with 1) first
From this last equation we have that x= 8 (mod 11) and x=3 (mod 11)
Now we will solve 2)
From this last equation we have that x=5 (mod 7) and x=2 (mod7)
Now we have 2 different pair of solutions, we're going to work with one equation modulo 11 and one modulo 7 at a time: because of the Chinese Remainder Theorem we know that each pair of equations has an answer mod 77.
Pair 1)
We rewrite the second equation as x = 7y +5, we substitute x in the first equation and we have:
7y + 5 = 8 (mod 11)
y = 2 (mod 11)
Now we rewrite this last equation as y = 11z + 2 and we will substitute this in x = 7y + 5
x = 7(11z + 2) + 5
x = 77z +14 + 5
x = 77 z + 19
x= 19 (mod 77)
Pair 2)
We will follow the same procedure:
x = 7y +2
7y + 2 = 8 (mod 11)
y = 4 (mod 11)
⇒y = 11z + 4
x = 7(11z + 4) + 2
x = 77z +28 + 2
x= 77z + 30
x = 30 (mod 77)
Pair 3)
x = 7y + 5
7y + 5 = 3 (mod 11)
y = 6 (mod 11)
⇒y =11z + 6
x = 7 (11z + 6) + 5
x = 77z + 42 + 5
x = 47 (mod 77)
Pair 4)
x = 7y + 2
7y + 2= 3 (mod 11)
y = 8 (mod 11)
⇒y = 11z + 8
x = 7 (11z + 8) + 2
x = 77z + 56 + 2
x = 58 (mod 77)