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.
To determine which statements of congruence are true and false, we need to apply the definition of congruence. Two integers "a" and "b" are said to be congruent modulo "m" if "(a - b)" is divisible by "m", denoted as "a ≡ b (mod m)".
Let's analyze each statement one by one:
1) 177 ≡ 17 (mod 8):
To check if this congruence is true, we subtract the second number (17) from the first number (177) and check if the difference is divisible by 8. In this case, (177 - 17) = 160, which is divisible by 8. Therefore, the congruence is true.
2) 871 ≡ 713 (mod 29):
Similarly, we subtract the second number (713) from the first number (871) and check if the difference is divisible by 29. In this case, (871 - 713) = 158, which is divisible by 29. Therefore, the congruence is true.
3) 1322 ≡ 5294 (mod 12):
Following the same steps, we find that (1322 - 5294) = -3972, which is not divisible by 12. Hence, the congruence is false.
4) 5141 ≡ 8353 (mod 11):
By subtracting the second number (8353) from the first number (5141), we get (5141 - 8353) = -3212, which is not divisible by 11. Therefore, the congruence is false.
5) 13944 ≡ 8919 (mod 13):
After subtracting the second number (8919) from the first number (13944), we obtain (13944 - 8919) = 5025, which is not divisible by 13. Thus, the congruence is false.
6) 67 × 73 ≡ 1 × 3 (mod 5):
Calculating the product on both sides, we have (67 × 73) ≡ (1 × 3) ≡ 3 (mod 5). Since both sides result in 3, the congruence is true.
7) 17 × 18 × 19 × 20 ≡ 4! (mod 8):
Expanding the product on the left side, we get (17 × 18 × 19 × 20) ≡ (4 × 3 × 2 × 1) ≡ 24 ≡ 0 (mod 8). Therefore, the congruence is true.
8) 83^(144) ≡ 15^(144) (mod 17):
To verify this congruence, we calculate the exponentiation on both sides. Computing (83^(144)) and (15^(144)), we find that both numbers are congruent to 1 (mod 17). Therefore, the congruence is true.
In summary:
True congruences:
- 177 ≡ 17 (mod 8)
- 871 ≡ 713 (mod 29)
- 67 × 73 ≡ 1 × 3 (mod 5)
- 17 × 18 × 19 × 20 ≡ 4! (mod 8)
- 83^(144) ≡ 15^(144) (mod 17)
False congruences:
- 1322 ≡ 5294 (mod 12)
- 5141 ≡ 8353 (mod 11)
- 13944 ≡ 8919 (mod 13)