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The least number,which when divided by 12,15,20 or 54,leaves a remainder of 4 in each case,is
- a)450
- b)454
- c)540
- d)544
Correct answer is option 'D'. Can you explain this answer?
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The least number,which when divided by 12,15,20 or 54,leaves a remaind...
Understanding the Problem
To find the least number that, when divided by 12, 15, 20, or 54, leaves a remainder of 4, we can set up the equation:
- Let the required number be x.
- Then, x can be expressed as:
x = k * LCM + 4
where LCM is the least common multiple of the given divisors and k is an integer.
Finding the LCM
First, we need to find the LCM of 12, 15, 20, and 54.
- Prime Factorizations:
- 12 = 2^2 * 3
- 15 = 3 * 5
- 20 = 2^2 * 5
- 54 = 2 * 3^3
- Calculating LCM:
- Take the highest powers of all prime factors:
- 2^2 (from 12 or 20)
- 3^3 (from 54)
- 5^1 (from 15 or 20)
- Therefore:
LCM = 2^2 * 3^3 * 5 = 4 * 27 * 5 = 540
Finding the Required Number
Now, we substitute back into our equation:
- x = k * 540 + 4
To find the least positive integer value for x, we set k = 1:
- x = 1 * 540 + 4 = 540 + 4 = 544
Conclusion
Thus, the least number which, when divided by 12, 15, 20, or 54, leaves a remainder of 4, is:
- 544
Therefore, the correct answer is option D.
To find the least number that, when divided by 12, 15, 20, or 54, leaves a remainder of 4, we can set up the equation:
- Let the required number be x.
- Then, x can be expressed as:
x = k * LCM + 4
where LCM is the least common multiple of the given divisors and k is an integer.
Finding the LCM
First, we need to find the LCM of 12, 15, 20, and 54.
- Prime Factorizations:
- 12 = 2^2 * 3
- 15 = 3 * 5
- 20 = 2^2 * 5
- 54 = 2 * 3^3
- Calculating LCM:
- Take the highest powers of all prime factors:
- 2^2 (from 12 or 20)
- 3^3 (from 54)
- 5^1 (from 15 or 20)
- Therefore:
LCM = 2^2 * 3^3 * 5 = 4 * 27 * 5 = 540
Finding the Required Number
Now, we substitute back into our equation:
- x = k * 540 + 4
To find the least positive integer value for x, we set k = 1:
- x = 1 * 540 + 4 = 540 + 4 = 544
Conclusion
Thus, the least number which, when divided by 12, 15, 20, or 54, leaves a remainder of 4, is:
- 544
Therefore, the correct answer is option D.
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The least number,which when divided by 12,15,20 or 54,leaves a remainder of 4 in each case,isa) 450 b) 454 c) 540 d) 544 Correct answer is option 'D'. Can you explain this answer?
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