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Sagot :
To determine the smallest and greatest digits that can replace '' in the given numbers such that the resulting number is divisible by 3, let's solve the problem step-by-step:
### Part (a) [tex]\( +672 \)[/tex]
1. Analyzing the problem:
- The number is in the form [tex]\( +672 \)[/tex].
- We need to determine the smallest and greatest digit that can replace '' to make the number divisible by 3.
2. Divisibility rule for 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Finding the smallest digit:
- Let's try replacing '' with digits from 0 to 9 and check which values make the number divisible by 3.
- We assume '+' is a single-digit number and try accordingly.
4. Finding the greatest digit:
- Similarly, check for the greatest possible value of '+' while ensuring that '' forms a number divisible by 3.
### Smallest digit:
After testing all possible digit replacements, the smallest digit that can replace '' to make [tex]\( +672 \)[/tex] divisible by 3 is [tex]\( 0 \)[/tex].
### Greatest digit:
The greatest digit that can replace '' to make [tex]\( +672 \)[/tex] divisible by 3 is [tex]\( 9 \)[/tex].
Therefore, for part (a), the smallest digit for '' is [tex]\( 0 \)[/tex], and the greatest digit is [tex]\( 9 \)[/tex].
### Part (b) [tex]\( 47562 \)[/tex]
1. Analyzing the problem:
- The number is in the form [tex]\( 47562 \)[/tex].
- We need to determine the smallest and greatest digit that can replace '' to make the number divisible by 3.
2. Divisibility rule for 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Finding the smallest digit:
- Let's try replacing '' with digits from 0 to 9 and check which values make the number divisible by 3.
4. Finding the greatest digit:
- Similarly, check for the digit that ensures the number remains divisible by 3.
### Smallest digit:
After testing all possible digit replacements, the smallest digit that can replace '' to make [tex]\(-+672 or 47562\)[/tex] divisible by 3 is [tex]\( 0 \)[/tex].
### Greatest digit:
After testing all possible digit replacements, the greatest digit that can replace '' to make [tex]\( 47562 \)[/tex] divisible by 3 is [tex]\( 9 \)[/tex].
Therefore, for part (b), the smallest digit for '' is [tex]\( 0 \)[/tex], and the greatest digit is [tex]\( 9 \)[/tex].
### Final Results:
- (a) For [tex]\( +672 \)[/tex]:
- The smallest digit is [tex]\( 0 \)[/tex].
- The greatest digit is [tex]\( 9 \)[/tex].
- (b) For [tex]\( 47562 \)[/tex]:
- The smallest digit is [tex]\( 0 \)[/tex].
- The greatest digit is [tex]\( 9 \)[/tex].
### Part (a) [tex]\( +672 \)[/tex]
1. Analyzing the problem:
- The number is in the form [tex]\( +672 \)[/tex].
- We need to determine the smallest and greatest digit that can replace '' to make the number divisible by 3.
2. Divisibility rule for 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Finding the smallest digit:
- Let's try replacing '' with digits from 0 to 9 and check which values make the number divisible by 3.
- We assume '+' is a single-digit number and try accordingly.
4. Finding the greatest digit:
- Similarly, check for the greatest possible value of '+' while ensuring that '' forms a number divisible by 3.
### Smallest digit:
After testing all possible digit replacements, the smallest digit that can replace '' to make [tex]\( +672 \)[/tex] divisible by 3 is [tex]\( 0 \)[/tex].
### Greatest digit:
The greatest digit that can replace '' to make [tex]\( +672 \)[/tex] divisible by 3 is [tex]\( 9 \)[/tex].
Therefore, for part (a), the smallest digit for '' is [tex]\( 0 \)[/tex], and the greatest digit is [tex]\( 9 \)[/tex].
### Part (b) [tex]\( 47562 \)[/tex]
1. Analyzing the problem:
- The number is in the form [tex]\( 47562 \)[/tex].
- We need to determine the smallest and greatest digit that can replace '' to make the number divisible by 3.
2. Divisibility rule for 3:
- A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Finding the smallest digit:
- Let's try replacing '' with digits from 0 to 9 and check which values make the number divisible by 3.
4. Finding the greatest digit:
- Similarly, check for the digit that ensures the number remains divisible by 3.
### Smallest digit:
After testing all possible digit replacements, the smallest digit that can replace '' to make [tex]\(-+672 or 47562\)[/tex] divisible by 3 is [tex]\( 0 \)[/tex].
### Greatest digit:
After testing all possible digit replacements, the greatest digit that can replace '' to make [tex]\( 47562 \)[/tex] divisible by 3 is [tex]\( 9 \)[/tex].
Therefore, for part (b), the smallest digit for '' is [tex]\( 0 \)[/tex], and the greatest digit is [tex]\( 9 \)[/tex].
### Final Results:
- (a) For [tex]\( +672 \)[/tex]:
- The smallest digit is [tex]\( 0 \)[/tex].
- The greatest digit is [tex]\( 9 \)[/tex].
- (b) For [tex]\( 47562 \)[/tex]:
- The smallest digit is [tex]\( 0 \)[/tex].
- The greatest digit is [tex]\( 9 \)[/tex].
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