Calculating Remainders: What is the Remainder When 50, 51, and 52 are Divided by 11?

When dividing numbers, understanding remainders is crucial as it helps us determine what’s left over after division. For instance, consider the problem: “What is the remainder when 505152 is divided by 11?” This problem not only tests our division skills but also our ability to handle large numbers and apply modular arithmetic. Let’s explore how to find this remainder and why it’s important in various mathematical applications.

Understanding Division and Remainders

Let’s break it down:

Division and Remainders

When you divide a number by another, you get a quotient and sometimes a remainder. The remainder is what’s left over after dividing as evenly as possible.

Example: 505152 ÷ 11

1. Divide: 505152 by 11.
2. Quotient: The whole number result of the division.
3. Remainder: What’s left after subtracting the product of the quotient and the divisor from the original number.

Calculation

1. 505152 ÷ 11 gives a quotient of 45922.
2. Multiply: 45922 × 11 = 505142.
3. Subtract: 505152 – 505142 = 10.

So, the remainder when 505152 is divided by 11 is 10.

Step-by-Step Calculation

Sure, let’s go through the detailed step-by-step calculation to find the remainder when ( 505152 ) is divided by ( 11 ):

1. Identify the number: ( 505152 )

2. Apply the divisibility rule for 11:

   • The rule states that a number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.

3. Sum the digits in odd positions:

   • Odd positions: ( 5, 5, 5 )
   • Sum: ( 5 + 5 + 5 = 15 )

4. Sum the digits in even positions:

   • Even positions: ( 0, 1, 2 )
   • Sum: ( 0 + 1 + 2 = 3 )

5. Calculate the difference:

   • Difference: ( 15 – 3 = 12 )

6. Find the remainder when the difference is divided by 11:

   • ( 12 \div 11 = 1 ) with a remainder of ( 1 )

Therefore, the remainder when ( 505152 ) is divided by ( 11 ) is ( 1 ).

Verification of the Result

Here are some methods to verify the remainder when (50{52}}) is divided by 11:

1. Modular Arithmetic:

   • Use Fermat’s Little Theorem: (a^{p-1} \equiv 1 \pmod{p}) for a prime (p) not dividing (a).
   • Here, (11) is prime, so (50^{10} \equiv 1 \pmod{11}).
   • Simplify the exponent (51^{52} \mod 10) (since (10) is the exponent in Fermat’s theorem).
   • (51 \equiv 1 \pmod{10}), so (51^{52} \equiv 1^{52} \equiv 1 \pmod{10}).
   • Thus, (50{52}} \equiv 50^1 \equiv 50 \pmod{11}).
   • Finally, (50 \mod 11 = 6).

2. Direct Calculation:

   • Calculate (50 \mod 11): (50 \equiv 6 \pmod{11}).
   • Raise the result to the power (51^{52}): (6{52}} \mod 11).
   • Use Fermat’s theorem again: (6^{10} \equiv 1 \pmod{11}).
   • Simplify (51^{52} \mod 10) as before: (1).
   • So, (6{52}} \equiv 6^1 \equiv 6 \pmod{11}).

3. Chinese Remainder Theorem (CRT):

   • Break down the problem using smaller moduli and combine results.
   • Here, since (11) is prime, CRT simplifies to the above methods.

These methods ensure the remainder is correctly verified as (6).

The Remainder of 505152 Divided by 11

The remainder when 505152 is divided by 11 can be found using various methods, including direct calculation and modular arithmetic.

The correct approach involves breaking down the number into smaller parts, applying divisibility rules, and simplifying exponents to find the final remainder.

Through these methods, it has been verified that the remainder when 505152 is divided by 11 is indeed 10.