To solve this problem, we need to find a number that satisfies a specific condition of fixed differences between the divisors and their respective remainders.
Step 1: Check the difference between each divisor and its remainder
Let's find the difference between each divisor and its corresponding remainder:
For 9: 9−4=5
For 10: 10−5=5
For 15: 15−10=5
For 20: 20−15=5
Since the difference is constant (5) in each case, the required number can be calculated using the formula:
Required Number=LCM(Divisors)−Constant Difference
Step 2: Find the LCM of the divisors
Now, we calculate the Least Common Multiple (LCM) of 9,10,15, and 20:
Prime factorization of 9=32
Prime factorization of 10=2×5
Prime factorization of 15=3×5
Prime factorization of 20=22×5
Taking the highest power of each prime factor involved:
LCM(9,10,15,20)=22×32×5
LCM(9,10,15,20)=4×9×5=180
Step 3: Calculate the required number
Subtract the constant difference (5) from the LCM obtained:
Required Number=180−5=175
Step 4: Check for multiples of LCM to match options
Since 175 is not in the options, the required smallest number matching the options must be of the form:
Number=(180×k)−5(where k is an integer)
Let's test values for k:
The value 355 matches Option C.
Correct Answer: C) 355
