To determine the digits that can be inserted into the blank space in the number [tex]\(8\_30\)[/tex] to make it divisible by 15, we follow these steps:
1. Understanding Divisibility by 15:
- A number must be divisible by both 3 and 5 to be divisible by 15.
2. Divisibility by 5:
- A number is divisible by 5 if its last digit is either 0 or 5.
3. Divisibility by 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
Given the number [tex]\(8\_30\)[/tex]:
1. Check for Divisibility by 5:
- Since the last digit is 0, the number will always be divisible by 5 regardless of the blank position.
2. Check for Divisibility by 3:
- We explore different digits (0 through 9) to replace the blank and check if the resulting number is divisible by 3.
We try each digit from 0 to 9 in place of the blank:
### Calculation:
- Insert 0: [tex]\(8030\)[/tex], sum of digits: [tex]\(8 + 0 + 3 + 0 = 11\)[/tex] (not divisible by 3)
- Insert 1: [tex]\(8130\)[/tex], sum of digits: [tex]\(8 + 1 + 3 + 0 = 12\)[/tex] (divisible by 3)
- Insert 2: [tex]\(8230\)[/tex], sum of digits: [tex]\(8 + 2 + 3 + 0 = 13\)[/tex] (not divisible by 3)
- Insert 3: [tex]\(8330\)[/tex], sum of digits: [tex]\(8 + 3 + 3 + 0 = 14\)[/tex] (not divisible by 3)
- Insert 4: [tex]\(8430\)[/tex], sum of digits: [tex]\(8 + 4 + 3 + 0 = 15\)[/tex] (divisible by 3)
- Insert 5: [tex]\(8530\)[/tex], sum of digits: [tex]\(8 + 5 + 3 + 0 = 16\)[/tex] (not divisible by 3)
- Insert 6: [tex]\(8630\)[/tex], sum of digits: [tex]\(8 + 6 + 3 + 0 = 17\)[/tex] (not divisible by 3)
- Insert 7: [tex]\(8730\)[/tex], sum of digits: [tex]\(8 + 7 + 3 + 0 = 18\)[/tex] (divisible by 3)
- Insert 8: [tex]\(8830\)[/tex], sum of digits: [tex]\(8 + 8 + 3 + 0 = 19\)[/tex] (not divisible by 3)
- Insert 9: [tex]\(8930\)[/tex], sum of digits: [tex]\(8 + 9 + 3 + 0 = 20\)[/tex] (not divisible by 3)
The digits that make the sum divisible by 3 are:
[tex]\[ 1, 4, 7 \][/tex]
### Answer:
All possible digits to replace the blank that make the number divisible by 15 are:
[tex]\[ 1, 4, 7 \][/tex]
