Divisibility Rules for 2, 4 and 10: Complete the missing numbers

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

• Understand the requirement for divisibility by 10.
• Apply the divisibility rule.
• Verify against the given choices.

Now, let's work through each step:
Step 1: The problem asks to complete the number $52\text{ }_—$ so that it is divisible by 10 without a remainder. This means we need the last digit to be 0.
Step 2: According to the divisibility rule for 10, a number must end in 0 to be divisible by 10. Therefore, the last digit must be 0.
Step 3: Among the choices provided, only choice 1 ($0$) makes $520$ divisible by 10, as $520 \div 10 = 52$ with a remainder of 0.

Therefore, the correct digit that completes the number as divisible by 10 is 0.

To determine the missing number that makes $13?$ divisible by 10, we must consider the divisibility rule for 10: A number is divisible by 10 if its last digit is 0. Therefore, for the number $13?$ to be divisible by 10, ? must be 0.

We examine the answer choices:

• Choice 1: 1. If ? is 1, the number $131$ is not divisible by 10.
• Choice 2: 2. If ? is 2, the number $132$ is not divisible by 10.
• Choice 3: 3. If ? is 3, the number $133$ is not divisible by 10.
• Choice 4: All answers are incorrect. The number should be $130$ to be divisible by 10.

Upon examining the choices, since none corresponds to a digit of 0, the correct answer is indicated by Choice 4: All answers are incorrect.

To solve this problem, we'll focus on ensuring divisibility by 10 for the given number $51\text{ }_{—\text{ —}}$.

The divisibility rule for 10 states that a number must end with the digit 0 to be divisible by 10.

Thus, the missing digits should form a number that ends in 0. Given the partial number $51\text{ }_{1\text{ —}}$, we need to check potential combinations that satisfy this rule.

• Evaluation of the required final digit: Only the digit 0 satisfies the condition for divisibility by 10, meaning the complete number should be $51y0$ where $y$ is any digit.

Hence, the missing sequence to ensure divisibility by 10 is $1, 0$. Therefore, $51\text{ }_{1\text{ }0}$ is the solution.

Returning to the choices provided, the correct answer is choice 1: $1, 0$.

To conclude, the number is completed as $5110$, which is divisible by 10.

To solve this problem, we need to ensure that the complete number is divisible by 2. According to the rules of divisibility for 2, a number is divisible by 2 if its last digit is even. Therefore, we need to select a digit to replace the blank that will result in an even number when appended to 214.

Let's evaluate the provided choices:

• Choice 1: 2144 – The number ends in 4, which is even. So, 2144 is divisible by 2.
• Choice 2: 2146 – The number ends in 6, which is even. So, 2146 is divisible by 2.
• Choice 3: 2148 – The number ends in 8, which is even. So, 2148 is divisible by 2.
• Choice 4: "All answers are correct" – Since all proposed numbers (2144, 2146, 2148) end in even digits, they are each divisible by 2.

Based on these evaluations, all individual choices result in a number that is divisible by 2.

Therefore, the correct answer is: All answers are correct.

To solve this problem, we need to apply the rule of divisibility by 2, which states that a number is divisible by 2 if its last digit is an even number.

Let's consider the possible digits for the underscore: $1$, $5$, $3$, and $8$.

• If we insert $1$, the number becomes $1051$, which ends in an odd digit, not divisible by 2.
• If we insert $5$, the number becomes $1055$, which ends in an odd digit, not divisible by 2.
• If we insert $3$, the number becomes $1053$, which ends in an odd digit, not divisible by 2.
• If we insert $8$, the number becomes $1058$, which ends in an even digit, divisible by 2.

Since the correct last digit that makes the number divisible by 2 is an even number, the solution to the problem is inserting $8$.

Therefore, the solution to the problem is $8$.

To solve this problem, we identify the numbers that can make $62\text{ }_{—}$ divisible by 2:

• A number is divisible by 2 if its last digit is even.
• In our choices, the even numbers are $0$, as it satisfies the divisibility rule for 2.

Substitute the even number:

• Try $0$: The number becomes $620$.
• $620 \div 2 = 310$, which is an integer showing $620$ is divisible by 2.

Therefore, the correct digit to ensure $62\text{ }_{—}$ is divisible by 2 is 0.

To solve this problem, we'll identify which digit can be placed in the blank space to make the number divisible by 2.

Step 1: Review the divisibility rule for 2.
A number is divisible by 2 if its last digit is even.

Step 2: Evaluate each possible choice for the blank in $512\_.$

• Choice 1: If we fill the blank with 1, the number becomes 5121, which ends with 1, an odd number. This is not divisible by 2.
• Choice 2: If we fill the blank with 2, the number becomes 5122, which ends with 2, an even number. This number is divisible by 2.
• Choice 3: If we fill the blank with 3, the number becomes 5123, which ends with 3, an odd number. This is not divisible by 2.
• Choice 4: If we fill the blank with 5, the number becomes 5125, which ends with 5, an odd number. This is not divisible by 2.

Conclusion: Among all choices, only when the blank is filled with 2, the resulting number 5122 is divisible by 2.

Therefore, the correct answer is $2$.

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

• Step 1: Identify the rule for divisibility by 4.

• Step 2: Apply this rule to the number $_{—\text{ }}2$.

• Step 3: Test each possible digit for the missing number to find a two-digit number divisible by 4.

Now, let's work through each step:
Step 1: The divisibility rule for 4 is that a number is divisible by 4 if the last two digits form a number divisible by 4.
Step 2: In the number $213\text{ }_{—\text{ }}2$, the last two digits are _{\—\text{ }}2. We want this two-digit number to be divisible by 4.
Step 3: Test the potential digits for blank using the choices given:

• If the blank is 0, the last two digits are 02, which is divisible by 4. But this isn't provided as a solution in the context.

• If the blank is 2, the last two digits are 22, which is not divisible by 4.

• If the blank is 3, the last two digits are 32, which is divisible by 4.

• If the blank is 4, the last two digits are 42, which is not divisible by 4.

Checking through these options shows that placing a 3 in the blank makes the number divisible by 4.

Therefore, the solution to the problem is $3$.

To solve this problem, we need to complete the number $54216\_$ so that the entire number is divisible by 4. The rule for divisibility by 4 is that the number formed by its last two digits must be divisible by 4.

Let's try each of the given possibilities for the missing digit:

• For 0: Make the number $542160$. Last two digits are 60. Check divisibility: $60 \div 4 = 15$, remainder 0.
• For 1: Make the number $542161$. Last two digits are 61. Check divisibility: $61 \div 4 = 15.25$, remainder is not 0.
• For 2: Make the number $542162$. Last two digits are 62. Check divisibility: $62 \div 4 = 15.5$, remainder is not 0.
• For 3: Make the number $542163$. Last two digits are 63. Check divisibility: $63 \div 4 = 15.75$, remainder is not 0.

Out of the available options, only appending 0 to make the number $542160$ satisfies the condition, as 60 is divisible by 4.

Therefore, the digit that should replace the missing underscore to make the number divisible by 4 is 0.

To solve the problem, we will ensure the two-digit number formed from the missing blanks at the end of 21 is divisible by 4. This requires performing the following steps:

• Step 1: Identify combinations for the blanks. Since this is a two-digit number, consider each combination within the range of numbers provided.
• Step 2: Check divisibility. Once the numbers are identified, check if they are divisible by 4.
• Step 3: Validate the correct number. The rule is that it must divide evenly with no remainder.

Numbers formed by potential combinations from the given choices are:
- $212$, $214$

Now, let's check each of these numbers:

• $212$: Dividing 212 by 4 gives $212 \div 4 = 53$. This is an integer, meaning 212 is divisible by 4.
• $214$: Dividing 214 by 4 gives $214 \div 4 = 53.5$. This is not an integer, hence 214 is not divisible by 4.

After evaluation, the number $212$ is divisible by 4.

The numbers that fill the blanks and ensure divisibility by 4 are 2, 4.