Which Statements of congruence are true and which are false and why?
177 _= 17 (mod 8)
871 _= 713 (mod 29)
1322 _= 5294 (mod 12)
5141 _= 8353 (mod 11)
13944 _= 8919 (mod 13)
67 x 73 _= 1 x 3 (mod 5)
17 x 18 x 19 x 20 _= 4! (mod 8)
83 (144 power) _= 15 (144 power) (mod 17)
Their not equals signs...they are three lines on top of one another.
177-17 = 160 = 20*8 true
871-713 = 158 false
1322-5294 = -3972 = -331*12 true
and so on
17*18*19*20/4! = 17*3*19*5 false
83 = 15 (mod 17), so true
To determine if the statements of congruence are true or false, we need to calculate the remainder (modulo) of both sides of the congruence equation.
In general, for an equation of the form a ≡ b (mod n), it means that "a" and "b" have the same remainder when divided by "n". If the remainders are the same, the congruence is true; otherwise, it is false.
Let's evaluate each statement:
1. 177 ≡ 17 (mod 8)
To calculate the modulo, divide 177 by 8: 177 ÷ 8 = 22 remainder 1.
Similarly, divide 17 by 8: 17 ÷ 8 = 2 remainder 1.
Since both sides have the same remainder of 1, the congruence is true.
2. 871 ≡ 713 (mod 29)
Divide 871 by 29: 871 ÷ 29 = 30 remainder 11.
Divide 713 by 29: 713 ÷ 29 = 24 remainder 17.
The remainders are different (11 and 17), so the congruence is false.
3. 1322 ≡ 5294 (mod 12)
Divide 1322 by 12: 1322 ÷ 12 = 110 remainder 2.
Divide 5294 by 12: 5294 ÷ 12 = 441 remainder 2.
Both sides have the same remainder of 2, so the congruence is true.
4. 5141 ≡ 8353 (mod 11)
Divide 5141 by 11: 5141 ÷ 11 = 467 remainder 4.
Divide 8353 by 11: 8353 ÷ 11 = 759 remainder 4.
Both sides have the same remainder of 4, so the congruence is true.
5. 13944 ≡ 8919 (mod 13)
Divide 13944 by 13: 13944 ÷ 13 = 1072 remainder 0.
Divide 8919 by 13: 8919 ÷ 13 = 685 remainder 4.
The remainders are different (0 and 4), so the congruence is false.
6. 67 x 73 ≡ 1 x 3 (mod 5)
Calculate the left side: 67 x 73 = 4889.
Calculate the right side: 1 x 3 = 3.
Divide both sides by 5: 4889 ÷ 5 = 977 remainder 4, 3 ÷ 5 = 0 remainder 3.
The remainders are different (4 and 3), so the congruence is false.
7. 17 x 18 x 19 x 20 ≡ 4! (mod 8)
Calculate the left side: 17 x 18 x 19 x 20 = 116,520.
Calculate the right side: 4! = 24.
Divide both sides by 8: 116,520 ÷ 8 = 14,565 remainder 0, 24 ÷ 8 = 3 remainder 0.
Both sides have the same remainder of 0, so the congruence is true.
8. 83^(144) ≡ 15^(144) (mod 17)
Calculating large exponents is computationally intensive. However, notice that 83 ≡ 15 (mod 17), which means they have the same remainder when divided by 17. Consequently, raising both sides to the same power will yield the same remainder when divided by 17. Therefore, the congruence is true.
To summarize:
True congruences: 177 ≡ 17 (mod 8), 1322 ≡ 5294 (mod 12), 5141 ≡ 8353 (mod 11), 17 x 18 x 19 x 20 ≡ 4! (mod 8), 83^(144) ≡ 15^(144) (mod 17).
False congruences: 871 ≡ 713 (mod 29), 13944 ≡ 8919 (mod 13), 67 x 73 ≡ 1 x 3 (mod 5).
Note: The three horizontal lines on top of one another represent congruence and are read as "is congruent to" or "equivalent to".