Solving the Divisibility Problem

The question asks us to identify which number from the given options divides the expression $4^{12} + 4^{13} + 4^{14} + 4^{15}$ completely.

Step-by-Step Solution

Step 1: Simplify the Expression

We start by simplifying the given expression: $4^{12} + 4^{13} + 4^{14} + 4^{15}$.

Notice that all terms have a common factor. The lowest power is $4^{12}$. We can factor this out:

$$4^{12} + 4^{13} + 4^{14} + 4^{15} = 4^{12}(1 + 4^1 + 4^2 + 4^3)$$

Step 2: Calculate the Sum in the Parenthesis

Now, let's calculate the value inside the parenthesis:

• $4^1 = 4$
• $4^2 = 16$
• $4^3 = 64$

So the sum is:

$$1 + 4 + 16 + 64 = 85$$

Substituting this back into our expression, we get:

$$4^{12} \times 85$$

Our task now is to find which of the options divides $4^{12} \times 85$.

Step 3: Check Divisibility by Each Option

Option 1: Checking Divisibility by 17

We need to see if 17 divides $4^{12} \times 85$.

Let's check if 17 divides 85.

$$85 \div 17 = 5$$

Since 17 is a factor of 85, the number 17 completely divides the expression $4^{12} \times 85$.

Option 2: Checking Divisibility by 11

We need to check if 11 divides $4^{12} \times 85$.

Neither $4^{12}$ nor 85 is divisible by 11.

• $4^{12}$ is a power of 4, and powers of 4 are $4, 16, 64, 256, \dots$. None of these are divisible by 11.
• $85 \div 11 = 7$ with a remainder of 8. So, 11 does not divide 85.

Since 11 divides neither factor, it cannot divide the product $4^{12} \times 85$.

Option 3: Checking Divisibility by 3

We need to check if 3 divides $4^{12} \times 85$.

We can use modular arithmetic. $4 \equiv 1 \pmod{3}$.

Therefore, $4^{12} \equiv 1^{12} \equiv 1 \pmod{3}$.

For 85, the sum of digits is $8+5=13$. Since 13 is not divisible by 3, 85 is not divisible by 3. $85 \equiv 1 \pmod{3}$.

So, $4^{12} \times 85 \equiv 1 \times 1 \equiv 1 \pmod{3}$.

Since the remainder is 1, the expression is not divisible by 3.

Option 4: Checking Divisibility by 7

We need to check if 7 divides $4^{12} \times 85$.

Let's look at the powers of 4 modulo 7:

• $4^1 \equiv 4 \pmod{7}$
• $4^2 \equiv 16 \equiv 2 \pmod{7}$
• $4^3 \equiv 4 \times 2 \equiv 8 \equiv 1 \pmod{7}$

Since $4^3 \equiv 1 \pmod{7}$, we can find $4^{12} \pmod{7}$:

$$4^{12} = (4^3)^4 \equiv 1^4 \equiv 1 \pmod{7}$$

Now let's check 85 modulo 7:

$$85 = 12 \times 7 + 1$$

So, $85 \equiv 1 \pmod{7}$.

Therefore, $4^{12} \times 85 \equiv 1 \times 1 \equiv 1 \pmod{7}$.

Since the remainder is 1, the expression is not divisible by 7.

Conclusion

Based on the divisibility checks, only the number 17 completely divides the expression $4^{12} + 4^{13} + 4^{14} + 4^{15}$.