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# MODULAR ARITHMETIC II

+2
187
5

Suppose you want to fill the 5x5 square below with the following restrictions:
(1) The numbers 1-5 must appear in every row and column;
(2) The sum of the numbers in gray boxes is 2 (mod 6). What number must go in the center square?

Oct 20, 2019

#1
+2

Don't care about the white squares, just care about the grey squares.

We can fill the horizontal grey row with 1 2 3 4 5

The remaining four grey squares will contain 4 numbers from the list 1 2 3 4 5

Since is 2 (mod 6)

we just guess and check

1  + 2  + 3 + 4 + 5 = 15

That is the sum of the center horizontal row.

We then can guess and check the sum of the vertical row plus 15 to see if it fits 2 mod ( 6)

Lets say the center square is 1?

Then 29 / 6 remainder is 5

center square is 2?

Then 28 / 6 remainder is 4

Center square is 3?

then 27/6 remainder is 3

Center square is 4?

then 26/6 remainder is 2

Center squar eis 5?

25/6 remainder is 1

We see by guess and checking that the answer is 4, four should go into the center square

Oct 20, 2019
#2
+2

Your better at modular arithemtic than you think

Oct 20, 2019
#3
+2

I actually used guess and check in this problem XD XD, I only understood the format meaning of modulus I actually barely even know how to apply it.

CalculatorUser  Oct 20, 2019
#4
+3

Suppose you want to fill the 5x5 square below with the following restrictions:
(1) The numbers 1-5 must appear in every row and column;
(2) The sum of the numbers in gray boxes is 2 (mod 6). What number must go in the center square?

Let the number go in the center square $$= \mathbf{x}$$

Let the sum of the numbers $$1+2+3+4+5 = 15$$

Let the sum of the numbers in gray boxes $$= 15 + (15-x)$$

Then:

$$\begin{array}{|rcll|} \hline 15 + (15-x) & \equiv & 2 \pmod 6 \\ 30-x & \equiv & 2 \pmod 6 \quad | \quad 30 \pmod 6 = 0 \\ 0-x & \equiv & 2 \pmod 6 \\ -x & \equiv & 2 \pmod 6 \quad | \quad \cdot (-1) \\ x & \equiv & -2 \pmod 6 \\ x & \equiv & -2+6 \pmod 6 \\ x & \equiv & 6-2 \pmod 6 \\ \mathbf{x} & \equiv & \mathbf{4 \pmod 6} \\ \hline \end{array}$$

The number 4 must go in the center square. Oct 21, 2019
#5
+1

Thank you for the modulus lesson!

CalculatorUser  Oct 24, 2019