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Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90?
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Find the smallest number which when decreased by 7 is exactly divisibl...
Understanding the Problem
To find the smallest number that, when decreased by 7, is divisible by 54, 60, and 90, we first need to identify the least common multiple (LCM) of these numbers.
Finding the LCM
1. Prime Factorization:
- 54 = 2 × 3^3
- 60 = 2^2 × 3 × 5
- 90 = 2 × 3^2 × 5
2. Identify the Highest Powers:
- From the prime factorization, we take the highest power of each prime:
- 2^2 (from 60)
- 3^3 (from 54)
- 5^1 (from 60 and 90)
3. Calculate the LCM:
\[
\text{LCM} = 2^2 × 3^3 × 5^1 = 4 × 27 × 5 = 540
\]
Finding the Smallest Number
Now, we know that our number, let’s call it \( N \), should satisfy the condition:
\[
N - 7 = 540k \quad (k \text{ is a positive integer})
\]
Rearranging gives us:
\[
N = 540k + 7
\]
To find the smallest \( N \), set \( k = 1 \):
\[
N = 540 × 1 + 7 = 547
\]
Conclusion
Thus, the smallest number which, when decreased by 7, is exactly divisible by 54, 60, and 90 is:
547
To find the smallest number that, when decreased by 7, is divisible by 54, 60, and 90, we first need to identify the least common multiple (LCM) of these numbers.
Finding the LCM
1. Prime Factorization:
- 54 = 2 × 3^3
- 60 = 2^2 × 3 × 5
- 90 = 2 × 3^2 × 5
2. Identify the Highest Powers:
- From the prime factorization, we take the highest power of each prime:
- 2^2 (from 60)
- 3^3 (from 54)
- 5^1 (from 60 and 90)
3. Calculate the LCM:
\[
\text{LCM} = 2^2 × 3^3 × 5^1 = 4 × 27 × 5 = 540
\]
Finding the Smallest Number
Now, we know that our number, let’s call it \( N \), should satisfy the condition:
\[
N - 7 = 540k \quad (k \text{ is a positive integer})
\]
Rearranging gives us:
\[
N = 540k + 7
\]
To find the smallest \( N \), set \( k = 1 \):
\[
N = 540 × 1 + 7 = 547
\]
Conclusion
Thus, the smallest number which, when decreased by 7, is exactly divisible by 54, 60, and 90 is:
547
Community Answer
Find the smallest number which when decreased by 7 is exactly divisibl...
533
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Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90?
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Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90? for Class 4 2024 is part of Class 4 preparation. The Question and answers have been prepared according to the Class 4 exam syllabus. Information about Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90? covers all topics & solutions for Class 4 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90?.
Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90? for Class 4 2024 is part of Class 4 preparation. The Question and answers have been prepared according to the Class 4 exam syllabus. Information about Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90? covers all topics & solutions for Class 4 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the smallest number which when decreased by 7 is exactly divisible by 54, 60 and 90?.
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