1. 2
2. 3
3. 37
Select the correct answer using the code given below:
Understanding the Divisibility Problem
We are asked to determine which numbers among 2, 3, and 37 can divide the expression \(555^{777} + 777^{555}\). We need to check the divisibility of this sum by each of these numbers separately.
Checking Divisibility by 2
To check for divisibility by 2, we look at the parity (odd or even) of the numbers involved.
- The number 555 is odd. Any positive integer power of an odd number is also odd. Therefore, \(555^{777}\) is an odd number.
- The number 777 is odd. Similarly, any positive integer power of an odd number is odd. Therefore, \(777^{555}\) is an odd number.
- The sum of two odd numbers is always an even number. (\(odd + odd = even\)).
Since \(555^{777}\) is odd and \(777^{555}\) is odd, their sum \(555^{777} + 777^{555}\) is even. An even number is always divisible by 2.
Conclusion: The expression is divisible by 2.
Checking Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. We can also use modular arithmetic.
- For 555: The sum of digits is \(5 + 5 + 5 = 15\). Since 15 is divisible by 3, 555 is divisible by 3. This means \(555 \equiv 0 \pmod{3}\).
- For 777: The sum of digits is \(7 + 7 + 7 = 21\). Since 21 is divisible by 3, 777 is divisible by 3. This means \(777 \equiv 0 \pmod{3}\).
Now consider the expression modulo 3:
- \(555^{777} \pmod{3}\): Since \(555 \equiv 0 \pmod{3}\), \(555^{777} \equiv 0^{777} \equiv 0 \pmod{3}\).
- \(777^{555} \pmod{3}\): Since \(777 \equiv 0 \pmod{3}\), \(777^{555} \equiv 0^{555} \equiv 0 \pmod{3}\).
Therefore, \(555^{777} + 777^{555} \equiv 0 + 0 \equiv 0 \pmod{3}\).
Conclusion: The expression is divisible by 3.
Checking Divisibility by 37
We check if the base numbers, 555 and 777, are divisible by 37.
- \(555 = 5 \times 111 = 5 \times (3 \times 37) = 15 \times 37\). So, 555 is divisible by 37. This means \(555 \equiv 0 \pmod{37}\).
- \(777 = 7 \times 111 = 7 \times (3 \times 37) = 21 \times 37\). So, 777 is divisible by 37. This means \(777 \equiv 0 \pmod{37}\).
Now consider the expression modulo 37:
- \(555^{777} \pmod{37}\): Since \(555 \equiv 0 \pmod{37}\), \(555^{777} \equiv 0^{777} \equiv 0 \pmod{37}\).
- \(777^{555} \pmod{37}\): Since \(777 \equiv 0 \pmod{37}\), \(777^{555} \equiv 0^{555} \equiv 0 \pmod{37}\).
Therefore, \(555^{777} + 777^{555} \equiv 0 + 0 \equiv 0 \pmod{37}\).
Conclusion: The expression is divisible by 37.
Final Answer Determination
Based on the checks above, the expression \(555^{777} + 777^{555}\) is divisible by:
- 2 (Yes)
- 3 (Yes)
- 37 (Yes)
This means the expression is divisible by all three numbers: 1, 2, and 3.
Referring to the options:
- Option 1: 1 and 2 only
- Option 2: 2 and 3 only
- Option 3: 1 and 3 only
- Option 4: 1, 2 and 3
The correct option is the one that includes all the divisors we found.
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?
- 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 ?- What is the remainder if we divide \(3^{10}\) by 7?
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:
