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A natural number, when divided by 3, 4, 6 and 7, leaves a remainder of 2 in each case. What is the smallest of all such numbers?
- a)213
- b)843
- c)86
- d)63
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A natural number, when divided by 3, 4, 6 and 7, leaves a remainder of...
⇒ LCM of 3, 4, 6 and 7 = 3 × 22 × 7 = 84
∴ The required number = 84 + 2 = 86
Most Upvoted Answer
A natural number, when divided by 3, 4, 6 and 7, leaves a remainder of...
Understanding the Problem
To find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 7, we can express this condition mathematically. If we denote the natural number as x, we have:
- x % 3 = 2
- x % 4 = 2
- x % 6 = 2
- x % 7 = 2
This means that if we subtract 2 from x, the resulting number must be divisible by each of these divisors.
Finding the Common Divisor
Let us denote y = x - 2. Therefore, we need y to be divisible by:
- 3
- 4
- 6
- 7
To find y, we must calculate the least common multiple (LCM) of these numbers.
Calculating LCM
- The prime factorization of the numbers:
- 3 = 3^1
- 4 = 2^2
- 6 = 2^1 * 3^1
- 7 = 7^1
- The LCM is found by taking the highest power of each prime:
- LCM = 2^2 * 3^1 * 7^1 = 4 * 3 * 7 = 84
So, y must be a multiple of 84.
Finding the Smallest Natural Number
Since y = x - 2, we can express x as:
x = y + 2
The smallest positive multiple of 84 is 84 itself.
Thus:
x = 84 + 2 = 86
Conclusion
The smallest natural number that satisfies the given conditions is 86, which corresponds to option 'C'.
To find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 7, we can express this condition mathematically. If we denote the natural number as x, we have:
- x % 3 = 2
- x % 4 = 2
- x % 6 = 2
- x % 7 = 2
This means that if we subtract 2 from x, the resulting number must be divisible by each of these divisors.
Finding the Common Divisor
Let us denote y = x - 2. Therefore, we need y to be divisible by:
- 3
- 4
- 6
- 7
To find y, we must calculate the least common multiple (LCM) of these numbers.
Calculating LCM
- The prime factorization of the numbers:
- 3 = 3^1
- 4 = 2^2
- 6 = 2^1 * 3^1
- 7 = 7^1
- The LCM is found by taking the highest power of each prime:
- LCM = 2^2 * 3^1 * 7^1 = 4 * 3 * 7 = 84
So, y must be a multiple of 84.
Finding the Smallest Natural Number
Since y = x - 2, we can express x as:
x = y + 2
The smallest positive multiple of 84 is 84 itself.
Thus:
x = 84 + 2 = 86
Conclusion
The smallest natural number that satisfies the given conditions is 86, which corresponds to option 'C'.
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A natural number, when divided by 3, 4, 6 and 7, leaves a remainder of 2 in each case. What is the smallest of all such numbers?a)213b)843c)86d)63Correct answer is option 'C'. Can you explain this answer?
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