What is the greatest integer among the following by which the number 5 5+ 7 5is divisible?
12
Understanding Divisibility and Powers
The problem asks us to find the greatest integer among the given options (6, 8, 11, 12) that divides the number $5^5 + 7^5$.
Analyzing the Expression $5^5 + 7^5$
The expression is in the form of a sum of powers, $a^n + b^n$. In this case, $a=5$, $b=7$, and $n=5$. There is a useful algebraic identity related to the sum of powers when the exponent is odd.
For any positive integer $n$ that is odd, the expression $a^n + b^n$ is always divisible by $(a+b)$.
In our problem, $n=5$, which is an odd number. So, $5^5 + 7^5$ must be divisible by $(5+7)$.
Let's calculate $(a+b)$:
Therefore, $5^5 + 7^5$ is divisible by 12.
Checking Divisibility with Given Options
Since $5^5 + 7^5$ is divisible by 12, let's examine the given options:
- Option 1: 6
If a number is divisible by 12, it is also divisible by any of its factors. The factors of 12 include 1, 2, 3, 4, 6, and 12. Since 6 is a factor of 12, $5^5 + 7^5$ is divisible by 6. - Option 2: 8
To check if $5^5 + 7^5$ is divisible by 8, we can calculate the value or use modular arithmetic. Let's calculate the value:$$5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$$ $$7^5 = 7 \times 7 \times 7 \times 7 \times 7 = 16807$$ $$5^5 + 7^5 = 3125 + 16807 = 19932$$To check divisibility by 8, we look at the last three digits of 19932, which form the number 932. Let's divide 932 by 8:$$\frac{932}{8} = \frac{800 + 132}{8} = \frac{800}{8} + \frac{132}{8} = 100 + 16 \text{ remainder } 4$$Since 932 is not divisible by 8, 19932 is not divisible by 8. Thus, $5^5 + 7^5$ is not divisible by 8. - Option 3: 11
To check if 19932 is divisible by 11, we can use the divisibility rule for 11. Sum the digits in the odd positions from the right, sum the digits in the even positions from the right, and find the difference. Sum of digits at odd positions (2nd, 4th, etc. from right): $2 + 9 + 1 = 12$ Sum of digits at even positions (1st, 3rd, etc. from right): $3 + 9 = 12$ Difference = $12 - 12 = 0$. Since the difference is 0, 19932 is divisible by 11. Thus, $5^5 + 7^5$ is divisible by 11. - Option 4: 12
As established using the property $a^n + b^n$ being divisible by $a+b$ for odd $n$, $5^5 + 7^5$ is divisible by 12. We can also verify this by dividing 19932 by 12:$$\frac{19932}{12} = 1661$$Since the result is an integer, 19932 is divisible by 12.
Comparing Divisors Among Options
Based on our checks, among the given options (6, 8, 11, 12), the number $5^5 + 7^5$ is divisible by 6, 11, and 12. It is not divisible by 8. We are looking for the greatest integer among these options by which the number is divisible. Comparing 6, 11, and 12, the greatest value is 12.
Final Answer Determination
The number $5^5 + 7^5$ is divisible by 6, 11, and 12 among the given options. The greatest of these is 12.
| Option | Is $5^5 + 7^5$ divisible? | Reason |
|---|---|---|
| 6 | Yes | 6 is a factor of 12, and $5^5 + 7^5$ is divisible by 12. |
| 8 | No | $19932 \div 8$ leaves a remainder. |
| 11 | Yes | $19932 \div 11 = 1812$ (using calculation error earlier; $19932/11 = 1812$ not 1661) - Let's recheck $19932/11$. $19932 = 11 \times 1812$. Yes, it is divisible. |
| 12 | Yes | $5^5 + 7^5$ is divisible by $5+7=12$ because the power is odd. Also $19932 \div 12 = 1661$. |
The options that divide $5^5 + 7^5$ are 6, 11, and 12. The greatest among these is 12.
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Divisibility Rules | Shortcuts to check if a number is divisible by another number without performing long division (e.g., rules for 8, 11, 12). |
| Sum of Powers $a^n + b^n$ | For positive integer $n$, if $n$ is odd, $a^n + b^n$ is divisible by $(a+b)$. |
| Factors and Multiples | If number A is divisible by number B, then B is a factor of A, and A is a multiple of B. If A is divisible by B, and B is divisible by C, then A is divisible by C. |
Additional Information: Related Number Properties
- Divisibility by 12: A number is divisible by 12 if it is divisible by both 3 and 4. For 19932:
- Divisibility by 4: Check the last two digits (32). Since 32 is divisible by 4 ($32 = 4 \times 8$), 19932 is divisible by 4.
- Divisibility by 3: Sum the digits ($1+9+9+3+2 = 24$). Since 24 is divisible by 3 ($24 = 3 \times 8$), 19932 is divisible by 3.
- Sum of powers $a^n - b^n$: For any positive integer $n$, $a^n - b^n$ is always divisible by $(a-b)$. If $n$ is even, $a^n - b^n$ is also divisible by $(a+b)$.
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