$-5 \equiv 63(\bmod 3)$
Answer (1)
Calculate − 5 ( mod 3 ) , which is 1 .
Calculate 63 ( mod 3 ) , which is 0 .
Compare the remainders: 1 = 0 .
Conclude that the congruence is False .
Explanation
Understanding the Congruence We are asked to verify the congruence − 5 ≡ 63 ( mod 3 ) . This means we need to check if − 5 and 63 have the same remainder when divided by 3 .
Calculating -5 mod 3 First, let's find the remainder of − 5 when divided by 3 . We can write − 5 = ( − 2 ) × 3 + 1 . So, − 5 ≡ 1 ( mod 3 ) .
Calculating 63 mod 3 Next, let's find the remainder of 63 when divided by 3 . Since 63 = 21 × 3 + 0 , we have 63 ≡ 0 ( mod 3 ) .
Comparing the Results Now we compare the remainders. We found that − 5 ≡ 1 ( mod 3 ) and 63 ≡ 0 ( mod 3 ) . Since 1 = 0 , the congruence − 5 ≡ 63 ( mod 3 ) is false.
Final Answer Therefore, the statement − 5 ≡ 63 ( mod 3 ) is false.
Examples
Modular arithmetic is used in cryptography to encrypt and decrypt messages. For example, the Caesar cipher, a simple substitution cipher, uses modular arithmetic to shift letters by a certain number of positions. If we represent letters as numbers (e.g., A=0, B=1, ..., Z=25), then encrypting a letter involves adding a key (the shift value) to the letter's number and taking the result modulo 26. Decrypting involves subtracting the key and taking the result modulo 26. This ensures that the result is always within the range of 0 to 25, corresponding to the letters of the alphabet.
