Finding the Remainder for a Large Concatenated Number
The question asks for the remainder when the number formed by concatenating integers from 1 to 100 (123456789101112...99100) is divided by 16.
Understanding the Divisibility Rule for 16
To find the remainder when a large number is divided by 16, we only need to consider the number formed by its last four digits. This is because 16 is a factor of \(10^4\) (10000). Specifically, \(10000 = 16 \times 625\). Therefore, any number can be written as \(N = 10000k + L\), where \(L\) is the number formed by the last four digits. Since \(10000k\) is perfectly divisible by 16, the remainder of \(N\) when divided by 16 is the same as the remainder of \(L\) when divided by 16.
Mathematically, if \(N\) is the large number, we are looking for \(N \pmod{16}\). We know that \(N = 10000k + L\). Since \(10000 \equiv 0 \pmod{16}\), then \(N \equiv 10000k + L \equiv 0 \times k + L \equiv L \pmod{16}\).
Identifying the Last Four Digits
The number is formed by writing the integers from 1 to 100 in sequence:
123456789101112... 9899100
The last few numbers concatenated are 98, 99, and 100.
The number ends with ...9899100.
The last four digits of this number are '9100'.
Calculating the Remainder
Now, we need to find the remainder when the number formed by the last four digits, which is 9100, is divided by 16.
We perform the division: \(9100 \div 16\).
- Divide 91 by 16: \(91 = 16 \times 5 + 11\). The quotient is 5, and the remainder is 11.
- Bring down the next digit (0) to form 110.
- Divide 110 by 16: \(110 = 16 \times 6 + 14\). The quotient is 6, and the remainder is 14.
- Bring down the next digit (0) to form 140.
- Divide 140 by 16: \(140 = 16 \times 8 + 12\). The quotient is 8, and the remainder is 12.
So, \(9100 = 16 \times 568 + 12\).
The remainder when 9100 is divided by 16 is 12.
Conclusion
According to the divisibility rule for 16, the remainder when the concatenated number (123...99100) is divided by 16 is the same as the remainder when its last four digits (9100) are divided by 16.
Therefore, the remainder is 12.
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: - 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:
