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What is the greatest number less than 1000 which when divided respectively by 5, 7 and 9 leaves the remainders 3, 5 and 7 respectively ?
Explanation
To solve this problem, we first determine the difference between each divisor and its corresponding remainder:
- 5 - 3 = 2
- 7 - 5 = 2
- 9 - 7 = 2
Since the difference (let's call it k) is constant (k = 2), the required number will follow the form: LCM(divisors) × n - k.
First, calculate the LCM of 5, 7, and 9. Since these numbers are co-prime, their LCM is 5 × 7 × 9 = 315.
The numbers satisfying the condition are of the form 315n - 2. We need the greatest such number less than 1000:
- If n = 1, number = 315(1) - 2 = 313
- If n = 2, number = 315(2) - 2 = 628
- If n = 3, number = 315(3) - 2 = 945 - 2 = 943
- If n = 4, number = 315(4) - 2 = 1258 (which exceeds 1000)
Therefore, the greatest number less than 1000 is 943.
SIMILAR QUESTIONS
Which one of the following is the largest 3-digit number which when divided by 12, 15 and 18 respectively, gives a remainder 5 in each case ?
A is the smallest positive integer which when divided by 9 and 12 leaves remainder 8. B is the smallest positive integer which when divided by 9 and 12 leaves remainder 5. Which one of the following is the value of A - B ?
The smallest number which when added to 10000 becomes divisible by 20, 24 and 30 is
What is the smallest number, which when multiplied by 9 gives the product having the digit 5 only in all places ?