If a number leaves a remainder of 1 when divided by 3 and a remainder of 2 when

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If a number leaves a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4, what is the smallest positive integer that satisfies these conditions?

Questions & Step-by-Step Solutions

If a number leaves a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4, what is the smallest positive integer that satisfies these conditions?

Step 1: Understand the problem. We need to find a number that gives a remainder of 1 when divided by 3 and a remainder of 2 when divided by 4.
Step 2: Write down the conditions mathematically. We can express the conditions as: Number % 3 = 1 and Number % 4 = 2.
Step 3: Start testing positive integers to find the smallest number that meets both conditions. Begin with 1 and go up.
Step 4: Check 1: 1 % 3 = 1 (good), 1 % 4 = 1 (not good).
Step 5: Check 2: 2 % 3 = 2 (not good), 2 % 4 = 2 (good).
Step 6: Check 3: 3 % 3 = 0 (not good), 3 % 4 = 3 (not good).
Step 7: Check 4: 4 % 3 = 1 (good), 4 % 4 = 0 (not good).
Step 8: Check 5: 5 % 3 = 2 (not good), 5 % 4 = 1 (not good).
Step 9: Check 6: 6 % 3 = 0 (not good), 6 % 4 = 2 (good).
Step 10: Check 7: 7 % 3 = 1 (good), 7 % 4 = 3 (not good).
Step 11: Check 8: 8 % 3 = 2 (not good), 8 % 4 = 0 (not good).
Step 12: Check 9: 9 % 3 = 0 (not good), 9 % 4 = 1 (not good).
Step 13: Check 10: 10 % 3 = 1 (good), 10 % 4 = 2 (good).
Step 14: Since 10 meets both conditions, we check if it's the smallest. The previous checks confirm it is.

Modular Arithmetic – Understanding how to work with remainders when dividing numbers.
System of Congruences – Solving multiple conditions that involve remainders.

If a number leaves a remainder of 1 when divided by 3 and a remainder of 2 when

Practice Questions

Questions & Step-by-Step Solutions

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