Which of the following numbers will completely divide 412 + 413 + 414 + 415?
17
Understanding the Problem: Divisibility of Powers
The question asks us to find which of the given options (3, 7, 11, or 17) completely divides the sum of four consecutive powers of 4, specifically $4^{12} + 4^{13} + 4^{14} + 4^{15}$. To solve this, we need to simplify the given expression and then check its divisibility by each option.
Simplifying the Expression $4^{12} + 4^{13} + 4^{14} + 4^{15}$
We can factor out the lowest power of 4, which is $4^{12}$, from each term in the sum. This is a common technique when dealing with sums of consecutive powers of the same base.
Let the expression be E:
E = $4^{12} + 4^{13} + 4^{14} + 4^{15}$
We can rewrite each term using the properties of exponents ($a^{m+n} = a^m \times a^n$):
- $4^{12} = 4^{12} \times 4^0$ (since $4^0 = 1$)
- $4^{13} = 4^{12} \times 4^1$
- $4^{14} = 4^{12} \times 4^2$
- $4^{15} = 4^{12} \times 4^3$
Now, substitute these back into the expression E:
E = $4^{12} \times 4^0 + 4^{12} \times 4^1 + 4^{12} \times 4^2 + 4^{12} \times 4^3$
Factor out $4^{12}$:
E = $4^{12} (4^0 + 4^1 + 4^2 + 4^3)$
Now, calculate the value inside the parenthesis:
- $4^0 = 1$
- $4^1 = 4$
- $4^2 = 4 \times 4 = 16$
- $4^3 = 4 \times 4 \times 4 = 64$
Summing these values:
$4^0 + 4^1 + 4^2 + 4^3 = 1 + 4 + 16 + 64$
$1 + 4 = 5$
$5 + 16 = 21$
$21 + 64 = 85$
So, the expression simplifies to:
E = $4^{12} \times 85$
Checking for Complete Divisibility
Now we need to determine which of the given options completely divides $4^{12} \times 85$. If a number divides either $4^{12}$ or 85 (or a combination of their factors), it will divide the product.
Let's look at the prime factorization of 85:
$85 = 5 \times 17$
The term $4^{12}$ has only prime factors of 2, since $4 = 2^2$, so $4^{12} = (2^2)^{12} = 2^{24}$.
So, the expression is $2^{24} \times 5 \times 17$.
Any number that completely divides this expression must be composed of the prime factors 2, 5, and 17, raised to powers no greater than their respective powers in the expression ($2^{24}, 5^1, 17^1$).
Comparing with the Options
Let's check each option:
- Option 1: 3
Is $2^{24} \times 5 \times 17$ divisible by 3? No, because neither $2^{24}$, 5, nor 17 have 3 as a prime factor. Therefore, the expression is not divisible by 3. - Option 2: 7
Is $2^{24} \times 5 \times 17$ divisible by 7? No, because neither $2^{24}$, 5, nor 17 have 7 as a prime factor. Therefore, the expression is not divisible by 7. - Option 3: 11
Is $2^{24} \times 5 \times 17$ divisible by 11? No, because neither $2^{24}$, 5, nor 17 have 11 as a prime factor. Therefore, the expression is not divisible by 11. - Option 4: 17
Is $2^{24} \times 5 \times 17$ divisible by 17? Yes, because 17 is one of the prime factors of the expression ($17^1$ is a factor). The expression can be written as $17 \times (2^{24} \times 5)$, which shows it is a multiple of 17. Therefore, the expression is completely divisible by 17.
Based on this analysis, the number that completely divides $4^{12} + 4^{13} + 4^{14} + 4^{15}$ is 17.
Conclusion
The simplified form of the expression $4^{12} + 4^{13} + 4^{14} + 4^{15}$ is $4^{12} \times 85$. Since $85 = 5 \times 17$, the expression is $4^{12} \times 5 \times 17$. A number completely divides this expression if it is a factor of $4^{12}$, 5, or 17, or a product of these factors. Among the given options (3, 7, 11, 17), only 17 is a factor of the expression $4^{12} \times 5 \times 17$.
| Expression | Simplified Form | Prime Factors | Divisible by 17? |
|---|---|---|---|
| $4^{12} + 4^{13} + 4^{14} + 4^{15}$ | $4^{12} \times 85$ | $2^{24} \times 5 \times 17$ | Yes |
Revision Table: Key Concepts
| Concept | Description | Relevance to Problem |
|---|---|---|
| Factoring Exponents | Extracting the lowest power from a sum of terms with the same base and different exponents, e.g., $a^m + a^{m+1} = a^m(1 + a)$. | Used to simplify $4^{12} + 4^{13} + 4^{14} + 4^{15}$. |
| Calculating Powers | Finding the value of a base raised to an exponent, e.g., $4^2 = 16$. | Used to evaluate $4^0, 4^1, 4^2, 4^3$. |
| Prime Factorization | Breaking down a composite number into its prime factors, e.g., $85 = 5 \times 17$. | Used to identify the fundamental components of 85 and the entire expression. |
| Divisibility Rules | Rules or methods to check if one number is divisible by another. A number is divisible by another if all prime factors of the divisor are present in the dividend with at least the same powers. | Used to check if $4^{12} \times 85$ is divisible by 3, 7, 11, and 17. |
Additional Information: Divisibility and Factors
Complete divisibility means that when you divide one number by another, the remainder is zero. In other words, the divisor is a factor of the dividend.
For a product of numbers, say $A \times B$, to be divisible by a number D, D must be a factor of A, a factor of B, or a factor formed by combining factors of A and B.
In our case, the expression is $4^{12} \times 85$. The divisors are 3, 7, 11, and 17.
- Since $4^{12}$ only has prime factors of 2, it is not divisible by 3, 7, 11, or 17.
- We check if 85 is divisible by the options. $85 = 5 \times 17$.
- 85 is not divisible by 3 (since 3 is not a factor of 85).
- 85 is not divisible by 7 (since 7 is not a factor of 85).
- 85 is not divisible by 11 (since 11 is not a factor of 85).
- 85 is divisible by 17 (since 17 is a factor of 85).
Since 17 divides 85, it follows that 17 completely divides the product $4^{12} \times 85$. This confirms our earlier finding.
Important Questions from Multiples and Factors
If 847 × 385 × 675 × 3025 = 3 a × 5 b × 7 c × 11 d, then the value of ab – cd is:
(mx + n) is a factor of:
If 7-digit number 678p37q is divisible by 75 and p is not a composite, then the values of p and q are:
Which of the following numbers Is divisible by 24?
Which number among 24963, 24973, 24983 and 24993 is divisible by 7?
