Answer :
Certainly! Let's determine whether each number ([tex]$135$[/tex], [tex]$32390$[/tex], and [tex]$98$[/tex]) is divisible by [tex]$2$[/tex], [tex]$3$[/tex], [tex]$5$[/tex], [tex]$6$[/tex], [tex]$9$[/tex], and [tex]$10$[/tex]. Once we finish checking, we'll place a box if it is divisible by the respective number, and a dollar sign if it is not.
### Checking Divisibility for [tex]$135$[/tex]
1. Divisibility by [tex]$2$[/tex]: [tex]$135$[/tex] is odd, hence not divisible by [tex]$2$[/tex].
2. Divisibility by [tex]$3$[/tex]: Sum of digits = [tex]$1 + 3 + 5 = 9$[/tex] (9 is divisible by 3), hence divisible by [tex]$3$[/tex].
3. Divisibility by [tex]$5$[/tex]: Last digit is [tex]$5$[/tex], hence divisible by [tex]$5$[/tex].
4. Divisibility by [tex]$6$[/tex]: Since [tex]$135$[/tex] is not divisible by [tex]$2$[/tex], it is not divisible by [tex]$6$[/tex].
5. Divisibility by [tex]$9$[/tex]: Sum of digits = [tex]$9$[/tex] which is divisible by [tex]$9$[/tex], hence divisible by [tex]$9$[/tex].
6. Divisibility by [tex]$10$[/tex]: Last digit is [tex]$5$[/tex], so not divisible by [tex]$10$[/tex].
### Results for [tex]$135$[/tex]
[tex]\[ \begin{array}{cccccc} \underline{135:} & \$ & \square & \square & \$ & \square & \$ \\ \end{array} \][/tex]
### Checking Divisibility for [tex]$32390$[/tex]
1. Divisibility by [tex]$2$[/tex]: Last digit is [tex]$0$[/tex] (even), hence divisible by [tex]$2$[/tex].
2. Divisibility by [tex]$3$[/tex]: Sum of digits = [tex]$3 + 2 + 3 + 9 + 0 = 17$[/tex] (17 is not divisible by 3), hence not divisible by [tex]$3$[/tex].
3. Divisibility by [tex]$5$[/tex]: Last digit is [tex]$0$[/tex], hence divisible by [tex]$5$[/tex].
4. Divisibility by [tex]$6$[/tex]: Though [tex]$32390$[/tex] is divisible by [tex]$2$[/tex], it is not divisble by [tex]$3$[/tex], so it is not divisible by [tex]$6$[/tex].
5. Divisibility by [tex]$9$[/tex]: Sum of the digits = [tex]$17$[/tex] which is not divisible by [tex]$9$[/tex], hence not divisible by [tex]$9$[/tex].
6. Divisibility by [tex]$10$[/tex]: Last digit is [tex]$0$[/tex], hence divisible by [tex]$10$[/tex].
### Results for [tex]$32390$[/tex]
[tex]\[ \begin{array}{cccccc} \underline{32390:} & \square & \$ & \square & \$ & \$ & \square \\ \end{array} \][/tex]
### Checking Divisibility for [tex]$98$[/tex]
1. Divisibility by [tex]$2$[/tex]: Last digit is [tex]$8$[/tex] (even), hence divisible by [tex]$2$[/tex].
2. Divisibility by [tex]$3$[/tex]: Sum of digits = [tex]$9 + 8 = 17$[/tex] (17 is not divisible by 3), hence not divisible by [tex]$3$[/tex].
3. Divisibility by [tex]$5$[/tex]: Last digit is [tex]$8$[/tex], hence not divisible by [tex]$5$[/tex].
4. Divisibility by [tex]$6$[/tex]: Though [tex]$98$[/tex] is divisible by [tex]$2$[/tex], it is not divisible by [tex]$3$[/tex], so it is not divisible by [tex]$6$[/tex].
5. Divisibility by [tex]$9$[/tex]: Sum of the digits = [tex]$17$[/tex] which is not divisible by [tex]$9$[/tex], hence not divisible by [tex]$9$[/tex].
6. Divisibility by [tex]$10$[/tex]: Last digit is [tex]$8$[/tex], hence not divisible by [tex]$10$[/tex].
### Results for [tex]$98$[/tex]
[tex]\[ \begin{array}{cccccc} \underline{98:} & \square & \$ & \$ & \$ & \$ & \$ \\ \end{array} \][/tex]
### Final Summary:
- For [tex]$135$[/tex]: [tex]\[ \$ \; \square \; \square \; \$ \; \square \; \$ \][/tex]
- For [tex]$32390$[/tex]: [tex]\[ \square \; \$ \; \square \; \$ \; \$ \; \square \][/tex]
- For [tex]$98$[/tex]: [tex]\[ \square \; \$ \; \$ \; \$ \; \$ \; \$ \][/tex]
So, the final boxed representation should be:
[tex]\[ \text{135:} \quad \$ \; \square \; \square \; \$ \; \square \; \$ \\ \text{32390:} \quad \square \; \$ \; \square \; \$ \; \$ \; \square \\ \text{98:} \quad \square \; \$ \; \$ \; \$ \; \$ \; \$ \\ \][/tex]
### Checking Divisibility for [tex]$135$[/tex]
1. Divisibility by [tex]$2$[/tex]: [tex]$135$[/tex] is odd, hence not divisible by [tex]$2$[/tex].
2. Divisibility by [tex]$3$[/tex]: Sum of digits = [tex]$1 + 3 + 5 = 9$[/tex] (9 is divisible by 3), hence divisible by [tex]$3$[/tex].
3. Divisibility by [tex]$5$[/tex]: Last digit is [tex]$5$[/tex], hence divisible by [tex]$5$[/tex].
4. Divisibility by [tex]$6$[/tex]: Since [tex]$135$[/tex] is not divisible by [tex]$2$[/tex], it is not divisible by [tex]$6$[/tex].
5. Divisibility by [tex]$9$[/tex]: Sum of digits = [tex]$9$[/tex] which is divisible by [tex]$9$[/tex], hence divisible by [tex]$9$[/tex].
6. Divisibility by [tex]$10$[/tex]: Last digit is [tex]$5$[/tex], so not divisible by [tex]$10$[/tex].
### Results for [tex]$135$[/tex]
[tex]\[ \begin{array}{cccccc} \underline{135:} & \$ & \square & \square & \$ & \square & \$ \\ \end{array} \][/tex]
### Checking Divisibility for [tex]$32390$[/tex]
1. Divisibility by [tex]$2$[/tex]: Last digit is [tex]$0$[/tex] (even), hence divisible by [tex]$2$[/tex].
2. Divisibility by [tex]$3$[/tex]: Sum of digits = [tex]$3 + 2 + 3 + 9 + 0 = 17$[/tex] (17 is not divisible by 3), hence not divisible by [tex]$3$[/tex].
3. Divisibility by [tex]$5$[/tex]: Last digit is [tex]$0$[/tex], hence divisible by [tex]$5$[/tex].
4. Divisibility by [tex]$6$[/tex]: Though [tex]$32390$[/tex] is divisible by [tex]$2$[/tex], it is not divisble by [tex]$3$[/tex], so it is not divisible by [tex]$6$[/tex].
5. Divisibility by [tex]$9$[/tex]: Sum of the digits = [tex]$17$[/tex] which is not divisible by [tex]$9$[/tex], hence not divisible by [tex]$9$[/tex].
6. Divisibility by [tex]$10$[/tex]: Last digit is [tex]$0$[/tex], hence divisible by [tex]$10$[/tex].
### Results for [tex]$32390$[/tex]
[tex]\[ \begin{array}{cccccc} \underline{32390:} & \square & \$ & \square & \$ & \$ & \square \\ \end{array} \][/tex]
### Checking Divisibility for [tex]$98$[/tex]
1. Divisibility by [tex]$2$[/tex]: Last digit is [tex]$8$[/tex] (even), hence divisible by [tex]$2$[/tex].
2. Divisibility by [tex]$3$[/tex]: Sum of digits = [tex]$9 + 8 = 17$[/tex] (17 is not divisible by 3), hence not divisible by [tex]$3$[/tex].
3. Divisibility by [tex]$5$[/tex]: Last digit is [tex]$8$[/tex], hence not divisible by [tex]$5$[/tex].
4. Divisibility by [tex]$6$[/tex]: Though [tex]$98$[/tex] is divisible by [tex]$2$[/tex], it is not divisible by [tex]$3$[/tex], so it is not divisible by [tex]$6$[/tex].
5. Divisibility by [tex]$9$[/tex]: Sum of the digits = [tex]$17$[/tex] which is not divisible by [tex]$9$[/tex], hence not divisible by [tex]$9$[/tex].
6. Divisibility by [tex]$10$[/tex]: Last digit is [tex]$8$[/tex], hence not divisible by [tex]$10$[/tex].
### Results for [tex]$98$[/tex]
[tex]\[ \begin{array}{cccccc} \underline{98:} & \square & \$ & \$ & \$ & \$ & \$ \\ \end{array} \][/tex]
### Final Summary:
- For [tex]$135$[/tex]: [tex]\[ \$ \; \square \; \square \; \$ \; \square \; \$ \][/tex]
- For [tex]$32390$[/tex]: [tex]\[ \square \; \$ \; \square \; \$ \; \$ \; \square \][/tex]
- For [tex]$98$[/tex]: [tex]\[ \square \; \$ \; \$ \; \$ \; \$ \; \$ \][/tex]
So, the final boxed representation should be:
[tex]\[ \text{135:} \quad \$ \; \square \; \square \; \$ \; \square \; \$ \\ \text{32390:} \quad \square \; \$ \; \square \; \$ \; \$ \; \square \\ \text{98:} \quad \square \; \$ \; \$ \; \$ \; \$ \; \$ \\ \][/tex]
