Answer :
Certainly! Let's walk through the process of finding whole numbers between 163 and 265 that are divisible by each of the given numbers: 2, 3, 4, 5, 6, 8, 9, 10, and 11.
### Step-by-Step Solution:
1. Find Numbers Divisible by 2:
A number is divisible by 2 if it is even. The even numbers between 163 and 265 are:
[tex]\[ \{164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260, 262, 264\} \][/tex]
2. Find Numbers Divisible by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3. The numbers between 163 and 265 divisible by 3 are:
[tex]\[ \{165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 210, 213, 216, 219, 222, 225, 228, 231, 234, 237, 240, 243, 246, 249, 252, 255, 258, 261, 264\} \][/tex]
3. Find Numbers Divisible by 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The numbers between 163 and 265 divisible by 4 are:
[tex]\[ \{164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264\} \][/tex]
4. Find Numbers Divisible by 5:
A number is divisible by 5 if it ends in 0 or 5. The numbers between 163 and 265 divisible by 5 are:
[tex]\[ \{165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265\} \][/tex]
5. Find Numbers Divisible by 6:
A number is divisible by 6 if it is divisible by both 2 and 3. The numbers between 163 and 265 divisible by 6 are:
[tex]\[ \{168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264\} \][/tex]
6. Find Numbers Divisible by 8:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. The numbers between 163 and 265 divisible by 8 are:
[tex]\[ \{168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264\} \][/tex]
7. Find Numbers Divisible by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9. The numbers between 163 and 265 divisible by 9 are:
[tex]\[ \{171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261\} \][/tex]
8. Find Numbers Divisible by 10:
A number is divisible by 10 if it ends in 0. The numbers between 163 and 265 divisible by 10 are:
[tex]\[ \{170, 180, 190, 200, 210, 220, 230, 240, 250, 260\} \][/tex]
9. Find Numbers Divisible by 11:
A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11. The numbers between 163 and 265 divisible by 11 are:
[tex]\[ \{165, 176, 187, 198, 209, 220, 231, 242, 253, 264\} \][/tex]
### Summary:
We have carefully examined each divisor and listed out the numbers in the given range that meet the criteria for divisibility. This approach ensures that you can apply similar techniques to solve any other range-based divisibility questions.
### Step-by-Step Solution:
1. Find Numbers Divisible by 2:
A number is divisible by 2 if it is even. The even numbers between 163 and 265 are:
[tex]\[ \{164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260, 262, 264\} \][/tex]
2. Find Numbers Divisible by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3. The numbers between 163 and 265 divisible by 3 are:
[tex]\[ \{165, 168, 171, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204, 207, 210, 213, 216, 219, 222, 225, 228, 231, 234, 237, 240, 243, 246, 249, 252, 255, 258, 261, 264\} \][/tex]
3. Find Numbers Divisible by 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The numbers between 163 and 265 divisible by 4 are:
[tex]\[ \{164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264\} \][/tex]
4. Find Numbers Divisible by 5:
A number is divisible by 5 if it ends in 0 or 5. The numbers between 163 and 265 divisible by 5 are:
[tex]\[ \{165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265\} \][/tex]
5. Find Numbers Divisible by 6:
A number is divisible by 6 if it is divisible by both 2 and 3. The numbers between 163 and 265 divisible by 6 are:
[tex]\[ \{168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264\} \][/tex]
6. Find Numbers Divisible by 8:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8. The numbers between 163 and 265 divisible by 8 are:
[tex]\[ \{168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264\} \][/tex]
7. Find Numbers Divisible by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9. The numbers between 163 and 265 divisible by 9 are:
[tex]\[ \{171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261\} \][/tex]
8. Find Numbers Divisible by 10:
A number is divisible by 10 if it ends in 0. The numbers between 163 and 265 divisible by 10 are:
[tex]\[ \{170, 180, 190, 200, 210, 220, 230, 240, 250, 260\} \][/tex]
9. Find Numbers Divisible by 11:
A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11. The numbers between 163 and 265 divisible by 11 are:
[tex]\[ \{165, 176, 187, 198, 209, 220, 231, 242, 253, 264\} \][/tex]
### Summary:
We have carefully examined each divisor and listed out the numbers in the given range that meet the criteria for divisibility. This approach ensures that you can apply similar techniques to solve any other range-based divisibility questions.
