To tackle this problem, we'll need to approach it step-by-step:
1. Identify the Least Common Multiple (LCM) of the divisors:
The divisors given are 36, 48, 49, 63, and 72. To find the smallest number that is divisible by all of these, we need to determine their Least Common Multiple (LCM).
2. Factorize each divisor into its prime factors:
- [tex]\( 36 = 2^2 \times 3^2 \)[/tex]
- [tex]\( 48 = 2^4 \times 3 \)[/tex]
- [tex]\( 49 = 7^2 \)[/tex]
- [tex]\( 63 = 3^2 \times 7 \)[/tex]
- [tex]\( 72 = 2^3 \times 3^2 \)[/tex]
3. Determine the LCM by taking the highest power of each prime number appearing in the factorization:
- For [tex]\( 2 \)[/tex], the highest power is [tex]\( 2^4 \)[/tex] (from 48).
- For [tex]\( 3 \)[/tex], the highest power is [tex]\( 3^2 \)[/tex] (from 36, 63, and 72).
- For [tex]\( 7 \)[/tex], the highest power is [tex]\( 7^2 \)[/tex] (from 49).
Therefore, the LCM is:
[tex]\[
\text{LCM} = 2^4 \times 3^2 \times 7^2
\][/tex]
4. Calculate the LCM:
[tex]\[
\text{LCM} = 16 \times 9 \times 49
\][/tex]
[tex]\[
\text{LCM} = 144 \times 49
\][/tex]
[tex]\[
\text{LCM} = 7056
\][/tex]
5. Determine the smallest 7-digit number and find the first 7-digit number that is divisible by the LCM:
The smallest 7-digit number is 1,000,000. To find the smallest 7-digit number divisible by 7056, divide 1,000,000 by 7056 and then round up to the nearest whole number.
Dividing:
[tex]\[
\frac{1,000,000}{7056} \approx 141.73
\][/tex]
The smallest whole number greater than 141.73 is 142. Therefore, multiply 7056 by 142 to get the smallest 7-digit number:
[tex]\[
7056 \times 142 = 1,001,952
\][/tex]
Thus, the LCM of the given divisors is 7056, and the smallest 7-digit number that is exactly divisible by all of them is 1,001,952.
