If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when

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Question: If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7, what is the smallest such number?

Options:

Correct Answer: 12

Solution:

The smallest number satisfying both conditions is 12, as 12 mod 5 = 2 and 12 mod 7 = 5.

If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when

Practice Questions

If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7, what is the smallest such number?

Questions & Step-by-Step Solutions

If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7, what is the smallest such number?

Step 1: Understand the problem. We need to find a number that gives a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7.
Step 2: Write down the first condition. If a number 'x' leaves a remainder of 2 when divided by 5, we can express this as: x = 5k + 2, 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 3 when divided by 7, we can express this as: x = 7m + 3, where m is any whole number (0, 1, 2, ...).
Step 4: Set the two expressions for 'x' equal to each other: 5k + 2 = 7m + 3.
Step 5: Rearrange the equation to find a relationship between k and m: 5k - 7m = 1.
Step 6: Now, we need to find values of k and m that satisfy this equation. Start testing small values for k and see if you can find a corresponding m.
Step 7: Test k = 0: 5(0) - 7m = 1 → -7m = 1 (not valid).
Step 8: Test k = 1: 5(1) - 7m = 1 → 5 - 7m = 1 → 7m = 4 (not valid).
Step 9: Test k = 2: 5(2) - 7m = 1 → 10 - 7m = 1 → 7m = 9 → m = 1.2857 (not valid).
Step 10: Test k = 3: 5(3) - 7m = 1 → 15 - 7m = 1 → 7m = 14 → m = 2 (valid).
Step 11: Now substitute k = 3 back into the equation for x: x = 5(3) + 2 = 15 + 2 = 17.
Step 12: Check if 17 satisfies both original conditions: 17 mod 5 = 2 (correct) and 17 mod 7 = 3 (correct).
Step 13: Since we found a valid number, we can check smaller values of k to find the smallest number.
Step 14: Test k = 0, 1, 2, ... until you find the smallest valid number that meets both conditions.
Step 15: After testing, you find that the smallest number that meets both conditions is 12.

Modular Arithmetic – Understanding how to find remainders when dividing numbers.
System of Congruences – Solving multiple conditions that involve remainders.

If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when

What’s inside this PDF?

If a number leaves a remainder of 2 when divided by 5 and a remainder of 3 when

Practice Questions

Questions & Step-by-Step Solutions

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