Find the largest 5 digit no which is completely divisible by
5, 21, 16, and 8.

Options

A. a)99120
B. b)99480
C. c)99630
D. d)99720

Answer

To find the largest 5-digit number that is completely divisible by 5, 21, 16, and 8, we can follow these steps:

### Step 1: Find the LCM of the given numbers
We need to find the Least Common Multiple (LCM) of the numbers 5, 21, 16, and 8.

- The prime factorization of the numbers is:
- 5 = 5^1
- 21 = 3^1 × 7^1
- 16 = 2^4
- 8 = 2^3

- To find the LCM, we take the highest power of each prime number:
- For 2, the highest power is 2^4 (from 16)
- For 3, the highest power is 3^1 (from 21)
- For 5, the highest power is 5^1 (from 5)
- For 7, the highest power is 7^1 (from 21)

- Therefore, the LCM is:
\[ LCM = 2^4 × 3^1 × 5^1 × 7^1 = 16 × 3 × 5 × 7 = 1680 \]

### Step 2: Find the largest 5-digit number
The largest 5-digit number is 99999.

### Step 3: Find the largest 5-digit number divisible by the LCM
To find the largest 5-digit number divisible by 1680, we divide 99999 by 1680 and take the floor of the result:
\[ \frac{99999}{1680} \approx 59.52 \]
Taking the floor gives us 59.

### Step 4: Calculate the largest number
Now, we multiply the LCM by 59 to find the largest 5-digit number that is divisible by 1680:
\[ 59 × 1680 = 99120 \]

### Step 5: Verify the result
To ensure that 99120 is divisible by 5, 21, 16, and 8:
- 99120 ÷ 5 = 19824 (divisible)
- 99120 ÷ 21 = 4720 (divisible)
- 99120 ÷ 16 = 6195 (divisible)
- 99120 ÷ 8 = 12390 (divisible)

Since all divisions yield whole numbers, 99120 is indeed divisible by all four numbers.

### Final Answer
The largest 5-digit number which is completely divisible by 5, 21, 16, and 8 is **99120**.

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