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

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Question: 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?

Options:

  1. 5
  2. 10
  3. 11
  4. 14

Correct Answer: 11

Solution: 

The smallest number that satisfies both conditions is 11 (11 % 3 = 2 and 11 % 4 = 3).

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

Practice Questions

Q1
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?
  1. 5
  2. 10
  3. 11
  4. 14

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.
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