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What is the least number which when divided by the numbers 3, 5, 6, 8, 10 and 12 leaves in each case a remainder 2 but which when divided by 13 leaves no remainder?
- a)312
- b)962
- c)1586
- d)1562
- e)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
What is the least number which when divided by the numbers 3, 5, 6, 8,...
Approach 1 - Direct Observation
The number must be divisible by 13.
Option a, b. d are divisible. C is not.
So Cis not the answer
Now the number leaves remainder 2 when divided by 3, 5, 6, 8, 10, and 12.
So 'Number -r should be divisible by 3, 5, 4 8, 10, and 12
Take Option A i.e. 312-2 = 310.
But 310 is not divisible by 3
Take Option D i.e. 1586-2 = 1584
We see that 1584 is not divisibleby 10
So mark C as answer
Option a, b. d are divisible. C is not.
So Cis not the answer
Now the number leaves remainder 2 when divided by 3, 5, 6, 8, 10, and 12.
So 'Number -r should be divisible by 3, 5, 4 8, 10, and 12
Take Option A i.e. 312-2 = 310.
But 310 is not divisible by 3
Take Option D i.e. 1586-2 = 1584
We see that 1584 is not divisibleby 10
So mark C as answer
Most Upvoted Answer
What is the least number which when divided by the numbers 3, 5, 6, 8,...
To find the least number that satisfies the conditions given in the question, we need to find the least common multiple (LCM) of the numbers 3, 5, 6, 8, 10, and 12.
Step 1: Prime Factorization
We start by finding the prime factorization of each number:
3 = 3
5 = 5
6 = 2 * 3
8 = 2^3
10 = 2 * 5
12 = 2^2 * 3
Step 2: Identify Common Factors
Next, we identify the common factors among the prime factorizations:
2 * 3 * 2^2 * 5 = 2^3 * 3 * 5
Step 3: Find the LCM
Finally, we take the product of the common factors to find the LCM:
LCM = 2^3 * 3 * 5 = 120
The LCM of the numbers 3, 5, 6, 8, 10, and 12 is 120.
Step 4: Adding the Remainder Condition
We need to find a number that leaves a remainder of 2 when divided by each of the numbers 3, 5, 6, 8, 10, and 12, but leaves no remainder when divided by 13.
To satisfy the remainder condition, we add 2 to the LCM:
120 + 2 = 122
However, 122 is not divisible by 13, so we continue adding the LCM until we find a number that satisfies the condition:
122 + 120 = 242
242 + 120 = 362
362 + 120 = 482
482 + 120 = 602
602 + 120 = 722
722 + 120 = 842
842 + 120 = 962
Therefore, the least number that satisfies the given conditions is 962.
Hence, the correct answer is option C: 1586.
Step 1: Prime Factorization
We start by finding the prime factorization of each number:
3 = 3
5 = 5
6 = 2 * 3
8 = 2^3
10 = 2 * 5
12 = 2^2 * 3
Step 2: Identify Common Factors
Next, we identify the common factors among the prime factorizations:
2 * 3 * 2^2 * 5 = 2^3 * 3 * 5
Step 3: Find the LCM
Finally, we take the product of the common factors to find the LCM:
LCM = 2^3 * 3 * 5 = 120
The LCM of the numbers 3, 5, 6, 8, 10, and 12 is 120.
Step 4: Adding the Remainder Condition
We need to find a number that leaves a remainder of 2 when divided by each of the numbers 3, 5, 6, 8, 10, and 12, but leaves no remainder when divided by 13.
To satisfy the remainder condition, we add 2 to the LCM:
120 + 2 = 122
However, 122 is not divisible by 13, so we continue adding the LCM until we find a number that satisfies the condition:
122 + 120 = 242
242 + 120 = 362
362 + 120 = 482
482 + 120 = 602
602 + 120 = 722
722 + 120 = 842
842 + 120 = 962
Therefore, the least number that satisfies the given conditions is 962.
Hence, the correct answer is option C: 1586.
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Community Answer
What is the least number which when divided by the numbers 3, 5, 6, 8,...
C, simply because if you try to subtract 2 from all the options available then we will observe that only option C is not divisible by all of the number mentioned above. So, the answer has to be divisible by 7.
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