Find all the square roots of x^2 = 53 (mod 77) by hand. 2 marks

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)

the equation for this situation is y=122+45. to solve the equation you would add 45 and 122 to get a total of 167lbs. in conclusion, grace's weight eight weeks ago was 167lbs.  

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

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