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In a nursery the care taker has to plant flowers in rows having equal numbers of flowers in each row. When 6,9,10, 12 flowers are planted in each row then 3 flowers get left unplanted. When flowers are planted in a row size of 11 none of the flowers are left. What could be the least number of such flowers. 
  • a)
    181
  • b)
    180
  • c)
    543
  • d)
    363
Correct answer is option 'D'. Can you explain this answer?
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In a nursery the care taker has to plant flowers in rows having equal ...
Understanding the Problem
To solve the problem, we need to find the least number of flowers that fulfills the given conditions regarding planting in rows.
Conditions Given
- When planting 6, 9, 10, or 12 flowers per row, 3 flowers are left unplanted.
- When planting in rows of 11, none are left unplanted.
Setting Up the Equations
1. Let the total number of flowers be \( N \).
2. According to the first condition:
\( N \equiv 3 \mod 6 \)
\( N \equiv 3 \mod 9 \)
\( N \equiv 3 \mod 10 \)
\( N \equiv 3 \mod 12 \)
This means \( N - 3 \) is divisible by 6, 9, 10, and 12.
3. According to the second condition:
\( N \equiv 0 \mod 11 \)
Finding the Least Common Multiple
To find \( N - 3 \), we can compute the least common multiple (LCM) of 6, 9, 10, and 12.
- The prime factorization is:
- 6 = \( 2 \times 3 \)
- 9 = \( 3^2 \)
- 10 = \( 2 \times 5 \)
- 12 = \( 2^2 \times 3 \)
- The LCM = \( 2^2 \times 3^2 \times 5 = 180 \).
Thus,
\( N - 3 = 180k \) for some integer \( k \), which leads to
\( N = 180k + 3 \).
Using the Second Condition
Now, substituting into the second condition:
\[ 180k + 3 \equiv 0 \mod 11 \]
This simplifies to:
\[ 180k \equiv -3 \mod 11 \]
Calculating \( 180 \mod 11 \):
\[ 180 \div 11 = 16 \text{ remainder } 4 \]
Thus,
\[ 180 \equiv 4 \mod 11 \]
This leads to:
\[ 4k \equiv -3 \mod 11 \]
or
\[ 4k \equiv 8 \mod 11 \]
Multiplying by the modular inverse of 4 (which is 3), we get:
\[ k \equiv 24 \mod 11 \]
So, \( k \equiv 2 \mod 11 \).
Finding the Least Value of N
Taking \( k = 2 \) gives:
\[ N = 180 \times 2 + 3 = 363 \].
Conclusion
Thus, the least number of flowers that meets all conditions is
363.
Therefore, the correct answer is option D.
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In a nursery the care taker has to plant flowers in rows having equal numbers of flowers in each row. When 6,9,10, 12 flowers are planted in each row then 3 flowers get left unplanted. When flowers are planted in a row size of 11 none of the flowers are left. What could be the least number of such flowers.a)181b)180c)543d)363Correct answer is option 'D'. Can you explain this answer?
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In a nursery the care taker has to plant flowers in rows having equal numbers of flowers in each row. When 6,9,10, 12 flowers are planted in each row then 3 flowers get left unplanted. When flowers are planted in a row size of 11 none of the flowers are left. What could be the least number of such flowers.a)181b)180c)543d)363Correct answer is option 'D'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about In a nursery the care taker has to plant flowers in rows having equal numbers of flowers in each row. When 6,9,10, 12 flowers are planted in each row then 3 flowers get left unplanted. When flowers are planted in a row size of 11 none of the flowers are left. What could be the least number of such flowers.a)181b)180c)543d)363Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a nursery the care taker has to plant flowers in rows having equal numbers of flowers in each row. When 6,9,10, 12 flowers are planted in each row then 3 flowers get left unplanted. When flowers are planted in a row size of 11 none of the flowers are left. What could be the least number of such flowers.a)181b)180c)543d)363Correct answer is option 'D'. Can you explain this answer?.
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