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If a number leaves a remainder of 1 when divided by 4 and a remainder of 2 when

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Question: If a number leaves a remainder of 1 when divided by 4 and a remainder of 2 when divided by 5, what is the smallest positive integer that satisfies these conditions?

Options:

  1. 6
  2. 9
  3. 11
  4. 14

Correct Answer: 6

Solution: 

The smallest number satisfying both conditions is 9, as 9 mod 4 = 1 and 9 mod 5 = 4.

If a number leaves a remainder of 1 when divided by 4 and a remainder of 2 when

Practice Questions

Q1
If a number leaves a remainder of 1 when divided by 4 and a remainder of 2 when divided by 5, what is the smallest positive integer that satisfies these conditions?
  1. 6
  2. 9
  3. 11
  4. 14

Questions & Step-by-Step Solutions

If a number leaves a remainder of 1 when divided by 4 and a remainder of 2 when divided by 5, 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 4 and a remainder of 2 when divided by 5.
  • Step 2: Write down the first condition. If a number 'x' leaves a remainder of 1 when divided by 4, we can express this as: x mod 4 = 1.
  • Step 3: Write down the second condition. If the same number 'x' leaves a remainder of 2 when divided by 5, we can express this as: x mod 5 = 2.
  • Step 4: List some numbers that satisfy the first condition (x mod 4 = 1). These numbers are: 1, 5, 9, 13, 17, ... (all numbers of the form 4k + 1 where k is a non-negative integer).
  • Step 5: Check each of these numbers against the second condition (x mod 5 = 2). Start with the smallest number from the list: 1, 5, 9, ...
  • Step 6: Check 1: 1 mod 5 = 1 (not 2). Check 5: 5 mod 5 = 0 (not 2). Check 9: 9 mod 5 = 4 (not 2). Check 13: 13 mod 5 = 3 (not 2). Check 17: 17 mod 5 = 2 (this works!).
  • Step 7: The smallest positive integer that satisfies both conditions is 17.
  • Modular Arithmetic – Understanding how to find remainders when dividing numbers.
  • System of Congruences – Solving multiple conditions that involve remainders.
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