The least number which when divided by 48, 64, 90, 120 will leave the remainders 38,54,80,110 respectively is

Options

A. 2870
B. 2860
C. 2890
D. 2880

Answer

To find the least number which, when divided by 48, 64, 90, and 120, leaves the remainders 38, 54, 80, and 110 respectively, we can follow these steps:

### Step 1: Calculate the differences between the divisors and the remainders.
We need to find the differences for each divisor and its corresponding remainder:
- For 48 and 38: \( 48 - 38 = 10 \)
- For 64 and 54: \( 64 - 54 = 10 \)
- For 90 and 80: \( 90 - 80 = 10 \)
- For 120 and 110: \( 120 - 110 = 10 \)

### Step 2: Identify that the differences are the same.
From the calculations, we see that the difference is consistently 10 for all cases. This means that the least number we are looking for will be 10 less than a multiple of the least common multiple (LCM) of the divisors.

### Step 3: Find the least common multiple (LCM) of the divisors.
To find the LCM of 48, 64, 90, and 120, we can use their prime factorization:
- \( 48 = 2^4 \times 3^1 \)
- \( 64 = 2^6 \)
- \( 90 = 2^1 \times 3^2 \times 5^1 \)
- \( 120 = 2^3 \times 3^1 \times 5^1 \)

Now, take the highest power of each prime:
- For \( 2 \): \( 2^6 \) (from 64)
- For \( 3 \): \( 3^2 \) (from 90)
- For \( 5 \): \( 5^1 \) (from 90 and 120)

Thus, the LCM is:
\[ LCM = 2^6 \times 3^2 \times 5^1 = 64 \times 9 \times 5 \]

### Step 4: Calculate the LCM.
Calculating this step by step:
- \( 64 \times 9 = 576 \)
- \( 576 \times 5 = 2880 \)

So, the LCM of 48, 64, 90, and 120 is 2880.

### Step 5: Subtract the difference from the LCM.
Now, we need to subtract the difference (which is 10) from the LCM:
\[ \text{Required number} = 2880 - 10 = 2870 \]

Thus, the least number which when divided by 48, 64, 90, and 120 leaves the remainders 38, 54, 80, and 110 respectively is **2870**.

### Final Answer:
2870

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