Consider the statement "If [math: x≡3(mod12)] and [math: y≡7(mod18)], then [math: x+y≡4(mod6)]". Which of the following is true?

Question

BlackTom AI · Accepted Answer

x = 12a + 3 and y = 18b + 7 for some integers a,b, so x + y = 12a + 18b + 10 = 6(2a + 3b + 1) + 4. Hence x + y ≡ 4 (mod 6).

1. Relevant knowledge points and concepts
- A congruence x ≡ r (mod m) means x = m·k + r for some integer k.
- If m divides n (here 6 divides 12 and 18), then a congruence modulo a multiple reduces to a congruence modulo the divisor by reducing the remainder modulo the smaller modulus.
- Congruences are preserved under addition.

2. Complete reasoning process step by step
- From x ≡ 3 (mod 12) write x = 12a + 3.
- From y ≡ 7 (mod 18) write y = 18b + 7.
- Add: x + y = (12a + 3) + (18b + 7) = 12a + 18b + 10.
- Factor 6: 12a + 18b + 10 = 6(2a + 3b + 1) + 4.

3. Intermediate calculation steps shown
- 12a + 18b + 10 = 6·(2a) + 6·(3b) + 6·1 + 4 = 6(2a+3b+1) + 4.
- Therefore (x + y) − 4 is divisible by 6.

4. Why the correct answer is correct
- The expression x+y always equals 6·(integer) + 4, so for every integers a,b (hence for every x,y satisfying the hypotheses) x + y ≡ 4 (mod 6). Equivalently, reduce remainders: 3 (mod 12) gives 3 (mod 6); 7 (mod 18) gives 1 (mod 6); 3+1 = 4 (mod 6).

5. Why the other options are incorrect
- "True for some but not for all": incorrect because the algebraic derivation shows the congruence holds for every x,y satisfying the given congruences, not merely some examples.
- "False for all": incorrect because explicit algebra shows it is always true; any counterexample attempt fails since the equation is algebraically forced.

6. Conclusion confirming the provided answer
- Therefore the statement is true for all integers x and y (matches the provided correct answer).

Consider the statement "If [math: x≡3(mod12)] and [math: y≡7(mod18)], then [math: x+y≡4(mod6)]". Which of the following is true?Single choice

Similar Questions

Is there an integer [math: z] such that [math: 504z≡7(mod385)] and [math: 0≤z<385]? If there is, enter such an integer [math: z]. If there isn't, enter NA.

Is there an integer [math: y] such that [math: 504y≡10(mod385)] and [math: 0≤y<385]? If there is, enter such an integer [math: y]. If there isn't, enter NA.

18+8%3-5/3

For integers a, b and k>0 which is NOT correct.

What is the value of: (2605321+4228) mod 13?

Select the correct value for 740 mod 2399. The following equalities may be useful: 72 mod 2399=49 74 mod 2399=2 78 mod 2399=4 716 mod 2399=16 732 mod 2399=256 764 mod 2399=512

What is the value of ((131)39+11⋅(−11)) mod 13?

What is the value of (260⋅5321+42⋅28) mod 13?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler