Understanding the Problem: Remainder Calculation
The question asks us to find the remainder when the number \(3^{10}\) is divided by 7. This is a problem in modular arithmetic, often expressed as finding the value of \(3^{10} \pmod{7}\).
Method 1: Finding a Pattern in Remainders
We can calculate the remainders of the first few powers of 3 when divided by 7 to see if there's a repeating pattern:
- \(3^1 \div 7\) has a remainder of 3. So, \(3^1 \equiv 3 \pmod{7}\).
- \(3^2 = 9\). \(9 \div 7\) has a remainder of 2. So, \(3^2 \equiv 2 \pmod{7}\).
- \(3^3 = 27\). \(27 \div 7\) has a remainder of 6 (\(27 = 3 \times 7 + 6\)). So, \(3^3 \equiv 6 \pmod{7}\).
- \(3^4 = 81\). \(81 \div 7\) has a remainder of 4 (\(81 = 11 \times 7 + 4\)). So, \(3^4 \equiv 4 \pmod{7}\).
- \(3^5 = 243\). \(243 \div 7\) has a remainder of 5 (\(243 = 34 \times 7 + 5\)). So, \(3^5 \equiv 5 \pmod{7}\).
- \(3^6 = 729\). \(729 \div 7\) has a remainder of 1 (\(729 = 104 \times 7 + 1\)). So, \(3^6 \equiv 1 \pmod{7}\).
Notice that once we reach a remainder of 1 (\(3^6 \equiv 1 \pmod{7}\)), the pattern of remainders will start repeating. The cycle of remainders is (3, 2, 6, 4, 5, 1), and the length of this cycle is 6.
Applying the Pattern to \(3^{10}\)
To find the remainder of \(3^{10}\) divided by 7, we need to figure out where in the cycle the 10th power falls. We do this by dividing the exponent (10) by the cycle length (6):
\(10 \div 6 = 1 \text{ \textbf{with a remainder of}} 4\)
This remainder of 4 tells us that \(3^{10}\) will have the same remainder as the 4th number in our cycle of remainders.
The 4th remainder in the cycle (3, 2, 6, 4, 5, 1) is 4.
Therefore, the remainder when \(3^{10}\) is divided by 7 is 4.
Method 2: Using Fermat's Little Theorem
Fermat's Little Theorem is a useful tool in modular arithmetic. It states that if '\(p\)' is a prime number, then for any integer '\(a\)' not divisible by '\(p\)', we have:
\(a^{p-1} \equiv 1 \pmod{p}\)
In our case, \(a=3\) and \(p=7\). Since 7 is prime and 3 is not divisible by 7, we can apply the theorem:
\(3^{7-1} \equiv 3^6 \equiv 1 \pmod{7}\)
Now we want to find \(3^{10} \pmod{7}\). We can rewrite the exponent 10 using the exponent from the theorem (6):
\(10 = 6 + 4\)
So, we can write \(3^{10}\) as:
\(3^{10} = 3^{6+4} = 3^6 \times 3^4\)
Now, let's find the remainder modulo 7:
\(3^{10} \equiv (3^6 \times 3^4) \pmod{7}\)
Since we know \(3^6 \equiv 1 \pmod{7}\), we substitute this in:
\(3^{10} \equiv (1 \times 3^4) \pmod{7}\)
\(3^{10} \equiv 3^4 \pmod{7}\)
Now, we just need to calculate \(3^4\) and find its remainder when divided by 7:
\(3^4 = 81\)
\(81 \div 7 = 11 \text{ \textbf{with a remainder of}} 4\)
So, \(3^4 \equiv 4 \pmod{7}\).
Putting it all together:
\(3^{10} \equiv 4 \pmod{7}\)
Both methods show that the remainder when \(3^{10}\) is divided by 7 is 4.
Similar Questions
- If \(n\) is a natural number, then what is the sum of all distinct remainders of \(4^n + 6^n + 9^n + 11^n\) when divided by 10 for various values of \(n\)?
- If \(a, b, c, d\) are natural numbers, then how many possible remainders are there when \(1^a + 2^b + 3^c + 4^d\) is divided by 10?
- The expression \(555^{777} + 777^{555}\) is divisible by which of the following?
1. 2
2. 3
3. 37
Select the correct answer using the code given below: - When the number (12345678910111213 ... 99100) is divided by 16, what will be the remainder?
- What is the smallest natural number \(n\) such that \((n + 1) \times n \times (n - 1) \times (n - 2) \times ... 3 \times 2 \times 1\) is divisible by 910?
- Consider the following statements :
1. If \((3m^3 + 2m^2 + 5m + n)/m\) is not an integer, where \(m\) and \(n\) are integers, then \(n\) is not divisible by \(m\).
2. \(5(8^m) + 2^{3m}\) is divisible by 48 for all whole numbers \(m\).
Which of the statements given above is/are correct? What is the remainder when \(111^{222} + 222^{333} + 333^{444}\) is divided by 5?
What is the remainder when \(3^{255}\) is divided by 28?
What is the value of \(x (0 \le x \le 8)\) if \((100^{97} + 100^{54} + x + 1)\) leaves a remainder 0 when divided by 9?
What is the remainder when
\(70 \times 71 \times 72 \times 73 \times 74 \times 75 \times 76 \times 77 \times 78 \times 79\) is divided by 1000 ?
Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
