A number leaves a remainder of 4 when divided by 9 and a remainder of 2 when div

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Question: A number leaves a remainder of 4 when divided by 9 and a remainder of 2 when divided by 5. What is the smallest such number? (2023)

Options:

Correct Answer: 23

Exam Year: 2023

Solution:

The smallest number that satisfies both conditions is 23 (23 % 9 = 5 and 23 % 5 = 3).

A number leaves a remainder of 4 when divided by 9 and a remainder of 2 when divided by 5. What is the smallest such number? (2023)

A number leaves a remainder of 4 when divided by 9 and a remainder of 2 when divided by 5. What is the smallest such number? (2023)

Step 1: Understand the problem. We need to find a number that gives a remainder of 4 when divided by 9 and a remainder of 2 when divided by 5.
Step 2: Write down the first condition. If a number leaves a remainder of 4 when divided by 9, we can express it as: number = 9k + 4, where k is a whole number (0, 1, 2, ...).
Step 3: Write down the second condition. If a number leaves a remainder of 2 when divided by 5, we can express it as: number = 5m + 2, where m is a whole number (0, 1, 2, ...).
Step 4: Set the two expressions equal to each other: 9k + 4 = 5m + 2.
Step 5: Rearrange the equation to find a relationship between k and m: 9k - 5m = -2.
Step 6: Start testing values for k to find a corresponding m that is a whole number.
Step 7: For k = 0: 9(0) - 5m = -2 → -5m = -2 → m = 0.4 (not a whole number).
Step 8: For k = 1: 9(1) - 5m = -2 → 9 - 5m = -2 → 5m = 11 → m = 2.2 (not a whole number).
Step 9: For k = 2: 9(2) - 5m = -2 → 18 - 5m = -2 → 5m = 20 → m = 4 (this is a whole number).
Step 10: Substitute k = 2 back into the first expression to find the number: number = 9(2) + 4 = 18 + 4 = 22.
Step 11: Check if 22 satisfies the second condition: 22 % 5 = 2 (this is correct).
Step 12: Check if 22 satisfies the first condition: 22 % 9 = 4 (this is correct).
Step 13: Since both conditions are satisfied, the smallest number is 22.

Modular Arithmetic – The problem involves finding a number that satisfies specific conditions related to remainders when divided by other numbers.
Chinese Remainder Theorem – The question can be approached using principles from the Chinese Remainder Theorem, which deals with solving systems of congruences.