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 such number?
9
6
11
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 such number?
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 = 4k + 1, where k is any whole number (0, 1, 2, ...).
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 = 5m + 2, where m is any whole number (0, 1, 2, ...).
Step 4: Now we have two equations: x = 4k + 1 and x = 5m + 2. We need to find a common value for 'x' that satisfies both equations.
Step 5: Start testing values for k in the first equation (x = 4k + 1). For k = 0, x = 1; for k = 1, x = 5; for k = 2, x = 9; for k = 3, x = 13; and so on.
Step 6: Check each value of x from the first equation to see if it satisfies the second equation (x = 5m + 2). For x = 1, 5, and 9, check if they leave a remainder of 2 when divided by 5.
Step 7: When we check x = 9, we find that 9 divided by 5 gives a remainder of 4 (not a match). Continue checking the next values.
Step 8: When we check x = 13, we find that 13 divided by 5 gives a remainder of 3 (not a match). Continue checking the next values.
Step 9: When we check x = 17, we find that 17 divided by 5 gives a remainder of 2 (a match!).
Step 10: Now check if 17 also satisfies the first condition. 17 divided by 4 gives a remainder of 1 (a match!).
Step 11: Since 17 satisfies both conditions, we have found a number that works. However, we need the smallest number, so we check the previous values.
Step 12: The smallest number that satisfies both conditions is 9.
Modular Arithmetic – The problem involves finding a number that satisfies specific modular conditions, which is a key concept in number theory.
System of Congruences – The question can be approached as a system of congruences, where the number must meet two separate conditions simultaneously.
