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 if the given statements of congruence are true or false, we need to apply the definition of congruence modulo n.
Statement 1: 177 ≡ 17 (mod 8)
To verify this statement, we need to check if the remainder of dividing both numbers by 8 is the same.
177 ÷ 8 = 22 remainder 1
17 ÷ 8 = 2 remainder 1
Both numbers have the same remainder (1) when divided by 8, so the statement is true.
Statement 2: 871 ≡ 713 (mod 29)
Similarly, we need to check if the remainder of dividing both numbers by 29 is the same.
871 ÷ 29 = 30 remainder 11
713 ÷ 29 = 24 remainder 17
The remainders are not equal (11 ≠ 17), so the statement is false.
Statement 3: 1322 ≡ 5294 (mod 12)
1322 ÷ 12 = 110 remainder 2
5294 ÷ 12 = 441 remainder 2
The remainders are equal (2 = 2), so the statement is true.
Statement 4: 5141 ≡ 8353 (mod 11)
5141 ÷ 11 = 467 remainder 4
8353 ÷ 11 = 759 remainder 4
The remainders are equal (4 = 4), so the statement is true.
Statement 5: 13944 ≡ 8919 (mod 13)
13944 ÷ 13 = 1072 remainder 0
8919 ÷ 13 = 686 remainder 11
The remainders are not equal (0 ≠ 11), so the statement is false.
Statement 6: 67 x 73 ≡ 1 x 3 (mod 5)
67 x 73 = 4889
1 x 3 = 3
4889 ÷ 5 = 977 remainder 4
3 ÷ 5 = 0 remainder 3
The remainders are not equal (4 ≠ 3), so the statement is false.
Statement 7: 17 x 18 x 19 x 20 ≡ 4! (mod 8)
17 x 18 x 19 x 20 = 116280
4! = 24
116280 ÷ 8 = 14535 remainder 0
24 ÷ 8 = 3 remainder 0
The remainders are equal (0 = 0), so the statement is true.
Statement 8: 83^144 ≡ 15^144 (mod 17)
To determine this congruence, we need to find the remainders of the numbers raised to the power of 144 modulo 17.
83^144 ≡ 15^144 ≡ 1 (mod 17)
Both numbers have the same remainder (1) when raised to the power of 144 modulo 17, so the statement is true.
In summary,
Statement 1: True
Statement 2: False
Statement 3: True
Statement 4: True
Statement 5: False
Statement 6: False
Statement 7: True
Statement 8: True