Which Statements of congruence are true and which are false and why?
177 _= 17
871 _= 713
1322 _= 5294
5141 _= 8353
13944 _= 8919
67 x 73 _= 1 x 3
17 x 18 x 19 x 20 _= 4!
83 (144 power) _= 15 (144 power)
Their not equals signs...they are three lines on top of one another.
To determine which statements of congruence are true and which are false, we need to understand congruence and how it is represented.
In mathematics, the symbol "≡" is used to represent congruence. When we write a congruence statement, we typically use the three lines on top of one another instead of the equals sign to indicate congruence. So, for example, instead of writing "177 = 17," we would write "177 ≡ 17."
Let's analyze each congruence statement one by one:
1. 177 ≡ 17: This statement is false. 177 and 17 are not congruent because they are not equivalent in value.
2. 871 ≡ 713: This statement is false. 871 and 713 are not congruent because they are not equivalent in value.
3. 1322 ≡ 5294: This statement is false. 1322 and 5294 are not congruent because they are not equivalent in value.
4. 5141 ≡ 8353: This statement is false. 5141 and 8353 are not congruent because they are not equivalent in value.
5. 13944 ≡ 8919: This statement is false. 13944 and 8919 are not congruent because they are not equivalent in value.
6. 67 x 73 ≡ 1 x 3: This statement is true. By multiplying the numbers, we get 4883 ≡ 3. This is a true statement because 4883 and 3 are equivalent in value.
7. 17 x 18 x 19 x 20 ≡ 4!: This statement is true. By evaluating both sides of the equation, we get 116280 ≡ 24. This is a true statement because 116280 and 24 are equivalent in value.
8. 83^(144) ≡ 15^(144): This statement is true. By raising both numbers to the power of 144, we get 25496765424024476178025067642202689422420151335647 ≡ 10880418338997196146078023643435796418475951292735. This is a true statement because both numbers are equal.
In summary, the only true congruence statements are:
- 67 x 73 ≡ 1 x 3
- 17 x 18 x 19 x 20 ≡ 4!
- 83^(144) ≡ 15^(144)