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?
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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.
