Let's solve these problems step by step.
### Part (a):
Find the smallest 4-digit number which is divisible by both 75 and 200:
1. First, we need to find the Least Common Multiple (LCM) of 75 and 200.
2. To find the LCM of two numbers, we use the formula:
[tex]\[
\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)}
\][/tex]
Here, [tex]\(a = 75\)[/tex] and [tex]\(b = 200\)[/tex].
3. Compute the Greatest Common Divisor (GCD) of 75 and 200. Using the properties of numbers:
[tex]\[
\text{GCD}(75, 200) = 25
\][/tex]
4. Substitute the values back into the LCM formula:
[tex]\[
\text{LCM}(75, 200) = \frac{|75 \cdot 200|}{25} = \frac{15000}{25} = 600
\][/tex]
5. The smallest 4-digit number is 1000. Now, we need to find the smallest multiple of 600 that is greater than or equal to 1000.
6. Divide 1000 by 600 and round up to the nearest whole number:
[tex]\[
\lceil \frac{1000}{600} \rceil = \lceil 1.67 \rceil = 2
\][/tex]
7. Multiply the LCM (600) by this rounded-up value:
[tex]\[
600 \times 2 = 1200
\][/tex]
Therefore, the smallest 4-digit number which is divisible by both 75 and 200 is 1200.
### Part (b):
Find the smallest 5-digit number which is exactly divisible by 192 and 256:
1. First, we need to find the Least Common Multiple (LCM) of 192 and 256.
2. Using the LCM formula:
[tex]\[
\text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)}
\][/tex]
Here, [tex]\(a = 192\)[/tex] and [tex]\(b = 256\)[/tex].
3. Compute the Greatest Common Divisor (GCD) of 192 and 256:
[tex]\[
\text{GCD}(192, 256) = 64
\][/tex]
4. Substitute the values into the LCM formula:
[tex]\[
\text{LCM}(192, 256) = \frac{|192 \cdot 256|}{64} = \frac{49152}{64} = 768
\][/tex]
5. The smallest 5-digit number is 10000. Now, we need to find the smallest multiple of 768 that is greater than or equal to 10000.
6. Divide 10000 by 768 and round up to the nearest whole number:
[tex]\[
\lceil \frac{10000}{768} \rceil = \lceil 13.02 \rceil = 14
\][/tex]
7. Multiply the LCM (768) by this rounded-up value:
[tex]\[
768 \times 14 = 10752
\][/tex]
Therefore, the smallest 5-digit number which is exactly divisible by both 192 and 256 is 10752.
Summarizing, the answers are:
- The smallest 4-digit number divisible by 75 and 200 is 1200.
- The smallest 5-digit number exactly divisible by 192 and 256 is 10752.
