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?
Practice Questions
1 question
Q1
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?
12
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The smallest number satisfying both conditions is 12, as 12 mod 5 = 2 and 12 mod 7 = 5.
Questions & Step-by-step Solutions
1 item
Q
Q: 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?
Solution: The smallest number satisfying both conditions is 12, as 12 mod 5 = 2 and 12 mod 7 = 5.
Steps: 15
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).
