Divisibility Rules for 3, 6 and 9: Complete the missing numbers

To solve this problem, we'll follow these steps:

• Step 1: Observe the given number $?321$ and identify the digits we have: 3, 2, and 1.
• Step 2: Sum these digits: $3 + 2 + 1 = 6$.
• Step 3: Add the digit represented by "?", which gives us $6 + ?$.
• Step 4: Find the digit "?" such that the sum $6 + ?$ is divisible by 3.

Now, let's proceed with the calculation:

We need to select a digit from 0 to 9 that makes $6 + ?$ a multiple of 3.

Checking each option:

• $? = 0$: $6 + 0 = 6$ (divisible by 3)
• $? = 1$: $6 + 1 = 7$ (not divisible by 3)
• $? = 2$: $6 + 2 = 8$ (not divisible by 3)
• $? = 3$: $6 + 3 = 9$ (divisible by 3)
• $? = 4$: $6 + 4 = 10$ (not divisible by 3)
• $? = 5$: $6 + 5 = 11$ (not divisible by 3)
• $? = 6$: $6 + 6 = 12$ (divisible by 3)
• $? = 7$: $6 + 7 = 13$ (not divisible by 3)
• $? = 8$: $6 + 8 = 14$ (not divisible by 3)
• $? = 9$: $6 + 9 = 15$ (divisible by 3)

Therefore, from the choices given, the correct digit is $\boldsymbol{9}$ since it is provided as an option.

Thus, the complete number that is divisible by 3 without a remainder is $\boldsymbol{9321}$.

To solve this problem, we'll use the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9.

• Step 1: Calculate the sum of the known digits.
  The number given is $6?22$, which means we have digits 6, ?, 2, and 2.
  Calculating the sum of known digits: $6 + 2 + 2 = 10$.
• Step 2: Add the missing digit $?$ and check for divisibility by 9.
  The total sum considering the missing digit will be $10 + ?$.
• Step 3: Determine which value of $?$ makes $10 + ?$ divisible by 9.
  We analyze the expression $10 + ? \equiv 0 \pmod{9}$.

Trying each option:

• If $? = 0$, $10 + 0 = 10$ (not divisible by 9).
• If $? = 1$, $10 + 1 = 11$ (not divisible by 9).
• If $? = 2$, $10 + 2 = 12$ (not divisible by 9).
• If $? = 3$, $10 + 3 = 13$ (not divisible by 9).
• If $? = 4$, $10 + 4 = 14$ (not divisible by 9).
• If $? = 5$, $10 + 5 = 15$ (not divisible by 9).
• If $? = 6$, $10 + 6 = 16$ (not divisible by 9).
• If $? = 7$, $10 + 7 = 17$ (not divisible by 9).
• If $? = 8$, $10 + 8 = 18$ (divisible by 9).
• If $? = 9$, $10 + 9 = 19$ (not divisible by 9).

Therefore, the correct digit to place in the question mark to make the number divisible by 9 is $8$.

The problem asks us to find the digit represented by '?' in the number $427?$ so that the entire number is divisible by 9.

To solve this problem, we will use the divisibility rule for 9, which states: A number is divisible by 9 if the sum of its digits is divisible by 9.

Let's calculate the sum of the known digits:

• The digits are 4, 2, and 7.
• Calculate the sum: $4 + 2 + 7 = 13$.

Now let's include the unknown digit, represented by $x$, in the sum. The total sum of the digits will be $S = 4 + 2 + 7 + x = 13 + x$.

We need $13 + x$ to be divisible by 9. So, find a value for $x$ such that $13 + x \equiv 0 \pmod{9}$.

Let's check each value for $x$ from 0 to 9:

• If $x = 0$, $13 + 0 = 13$, not divisible by 9.
• If $x = 1$, $13 + 1 = 14$, not divisible by 9.
• If $x = 2$, $13 + 2 = 15$, not divisible by 9.
• If $x = 3$, $13 + 3 = 16$, not divisible by 9.
• If $x = 4$, $13 + 4 = 17$, not divisible by 9.
• If $x = 5$, $13 + 5 = 18$, which is divisible by 9.

Thus, the sum of the digits is divisible by 9 when $x = 5$.

The correct digit to complete the number such that it is divisible by 9 is $5$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the sum of the known digits $6 + 3 + 2 = 11$.

• Step 2: Consider each choice for the missing digit and calculate the new sum:

  • $▯ = 9 : \quad 11 + 9 = 20$ (not divisible by 3)

  • $▯=8:\quad11+8=19$ (not divisible by 3)

  • $▯=7:\quad11+7=18$ (divisible by 3)

  • $▯=5:\quad11+5=16$ (not divisible by 3)

Check which sum is divisible by 3. The sum of 18 (when ▯ is replaced by 7) is divisible by 3.

Hence, the choice $\boxed{7}$ completes the number so it is divisible by 3 without a remainder.

Therefore, the correct digit to complete the number is $7$.

To solve this problem, we'll follow these steps:

• Step 1: Add the known digits of the number.
• Step 2: Determine which missing digit results in a total sum divisible by 3.
• Step 3: Validate the result against given choices.

Now, let's execute these steps:

Step 1: Calculate the sum of the known digits.

The given number format is $3\_52$. The known digits are 3, 5, and 2. Calculate their sum: $3 + 5 + 2 = 10$.

Step 2: Determine the missing digit that completes the sum to be divisible by 3.

Let's denote the missing digit as $x$. The total sum of the digits is $10 + x$. For the number to be divisible by 3, $10 + x$ must also be divisible by 3.

Examine the possible values for $x$:

• If $x = 0$, then $10 + 0 = 10$ (not divisible by 3).
• If $x = 1$, then $10 + 1 = 11$ (not divisible by 3).
• If $x = 2$, then $10 + 2 = 12$ (divisible by 3).
• If $x = 3$, then $10 + 3 = 13$ (not divisible by 3).
• Continue in this manner for $x = 4$ to $9$.

The correct $x$ that results in a sum divisible by 3 is 2.

Step 3: Validate this result against the provided choices.

Among the choices given, $2$ is indeed an option. Thus, the correct missing digit is $2$.

Therefore, the solution to the problem is that the missing digit $x$ is $\boxed{2}$.

To determine the digits needed in the number $3?8?$ so that it is divisible by 9, follow these steps:

• Step 1: According to the divisibility rule for 9, the sum of the digits in the number must be divisible by 9.
• Step 2: Compute the sum of known digits: $3 + 8 = 11$.
• Step 3: Let the unknown digits be represented by $x$ and $y$. The sum of all digits will be $11 + x + y$.
• Step 4: For the number to be divisible by 9, $11 + x + y$ must be a multiple of 9.
• Step 5: Determine possible values for $x$ and $y$ so that the sum is as follows: 11 + x + y = 18 (next multiple of 9 after 11).
• Step 6: Doing the arithmetic, we require $x + y = 7$.
• Step 7: Examine the possible digit combinations for $x$ and $y$ that sum to 7:
  - $(x, y) = (0, 7), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0)$.
• Step 8: Check given answer choices for these pairs and notice none match exactly.

Upon verifying these computations against the choices available, we conclude that the correct answer is None of the above.

To solve this problem, we need to find digits to replace the question marks in $54??$ such that the resulting number is divisible by 6. We'll use the divisibility rules for both 2 and 3.

First, since the number must be even, the last digit must be one of the even numbers: 0, 2, 4, 6, or 8.

Second, we need the sum of the digits to be divisible by 3. The initial digits '54' have a sum of $5 + 4 = 9$. Since 9 is already divisible by 3, we need the sum of the missing digits (let’s denote them as x and y) to also result in a number divisible by 3 when added to 9.

Therefore, we need $9 + x + y \equiv 0 \pmod{3}$. This simplifies to $x + y \equiv 0 \pmod{3}$.

Let us check with the provided options:

• Option 1: $3,3$: The sum $x + y = 3 + 3 = 6$. Divisible by 3. Last digit is 3 (odd), this does not satisfy divisibility by 2.
• Option 2: $6,8$: The sum $x + y = 6 + 8 = 14$. Not divisible by 3.
• Option 3: $5,5$: The sum $x + y = 5 + 5 = 10$. Not divisible by 3.
• Option 4: $3,1$: The sum $x + y = 3 + 1 = 4$. Not divisible by 3.

Since none of the provided options satisfy both divisibility conditions, the correct answer is "None of the above".

In conclusion, Option 5: "None of the above" is the correct choice.

To solve this problem, let's begin by considering the divisibility rules for 6:

• For a number to be divisible by 6, it must be divisible by both 2 and 3.

First, we analyze divisibility by 2:

• The last digit of the number (represented by '?') must be even. Possible even digits are 0, 2, 4, 6, 8.

Next, consider the divisibility by 3:

• The sum of all the digits, $5 + 4 + x + y$, must be divisible by 3.
• Given that $5 + 4 = 9$, we seek values for $x + y$ such that $9 + x + y$ is divisible by 3.

Now, let’s test possible values for $x$ (tens place) and $y$ (units place) using the conditions above:

• If $y = 0$, $x + 0$ must satisfy $9 + x + 0$ is divisible by 3 (i.e., $x + 9 \equiv 0 \mod 3$). Testing values, there’s no $x$ value making it divisible by 6.
• If $y = 2$, $x + 2$ must satisfy $9 + x + 2 \equiv 0 \mod 3$. It works when $x = 4$ because $9 + 4 + 2 = 15$ is divisible by 3.
• If $y = 4$, $x + 4$ must satisfy $9 + x + 4 \equiv 0 \mod 3$. No values of $x$ make this divisible by 6.

Therefore, the combination that satisfies both divisibility rules is $x = 4$ and $y = 2$.

Conclusion: The missing numbers making 54?? divisible by 6 are $4$ and $2$, matching the selection $4,2$ from the choices.