What is the remainder when the product of 335, 608 and 853 is divided by 13?
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Finding the Remainder of a Product When Divided by 13
The question asks for the remainder when the product of 335, 608, and 853 is divided by 13. To solve this problem, we can use the properties of modular arithmetic. A key property states that the remainder of a product is the same as the remainder of the product of the individual remainders.
Mathematically, if $a \equiv r_1 \pmod{n}$ and $b \equiv r_2 \pmod{n}$, then $a \times b \equiv r_1 \times r_2 \pmod{n}$. We can extend this property to the product of three or more numbers.
Step-by-Step Calculation of Remainders
We will first find the remainder of each number when divided by 13.
- Finding the Remainder of 335 divided by 13: To find the remainder of 335 when divided by 13, we perform division: $$335 \div 13$$ We find that $13 \times 20 = 260$ and $13 \times 5 = 65$. So, $13 \times 25 = 260 + 65 = 325$. The difference is $335 - 325 = 10$. Thus, $335 = 13 \times 25 + 10$. The remainder when 335 is divided by 13 is 10. In modular arithmetic notation, $335 \equiv 10 \pmod{13}$.
- Finding the Remainder of 608 divided by 13: Next, we find the remainder of 608 when divided by 13: $$608 \div 13$$ We can estimate $13 \times 40 = 520$. The difference is $608 - 520 = 88$. $13 \times 6 = 78$. So, $608 = 13 \times 40 + 88 = 13 \times 40 + 13 \times 6 + 10 = 13 \times (40 + 6) + 10 = 13 \times 46 + 10$. The remainder when 608 is divided by 13 is 10. In modular arithmetic notation, $608 \equiv 10 \pmod{13}$.
- Finding the Remainder of 853 divided by 13: Finally, we find the remainder of 853 when divided by 13: $$853 \div 13$$ We can estimate $13 \times 60 = 780$. The difference is $853 - 780 = 73$. $13 \times 5 = 65$. So, $853 = 13 \times 60 + 73 = 13 \times 60 + 13 \times 5 + 8 = 13 \times (60 + 5) + 8 = 13 \times 65 + 8$. The remainder when 853 is divided by 13 is 8. In modular arithmetic notation, $853 \equiv 8 \pmod{13}$.
Calculating the Remainder of the Product
Now, we multiply the individual remainders we found and then find the remainder of this product when divided by 13.
The remainders are 10, 10, and 8. Product of remainders = $10 \times 10 \times 8 = 100 \times 8 = 800$.
Now we find the remainder when 800 is divided by 13: $$800 \div 13$$ We can estimate $13 \times 60 = 780$. The difference is $800 - 780 = 20$. $13 \times 1 = 13$. The difference is $20 - 13 = 7$. So, $800 = 13 \times 60 + 20 = 13 \times 60 + 13 \times 1 + 7 = 13 \times (60 + 1) + 7 = 13 \times 61 + 7$. The remainder when 800 is divided by 13 is 7.
Therefore, the remainder when the product of 335, 608, and 853 is divided by 13 is 7.
| Number | Division by 13 | Remainder (modulo 13) |
|---|---|---|
| 335 | $335 = 13 \times 25 + 10$ | $335 \equiv 10 \pmod{13}$ |
| 608 | $608 = 13 \times 46 + 10$ | $608 \equiv 10 \pmod{13}$ |
| 853 | $853 = 13 \times 65 + 8$ | $853 \equiv 8 \pmod{13}$ |
Final Remainder Calculation
Product of remainders: $10 \times 10 \times 8 = 800$. Remainder of product of remainders when divided by 13: $800 \div 13$, Remainder is 7. $800 \equiv 7 \pmod{13}$.
So, the remainder when the product $335 \times 608 \times 853$ is divided by 13 is 7. $335 \times 608 \times 853 \equiv 10 \times 10 \times 8 \pmod{13}$ $335 \times 608 \times 853 \equiv 800 \pmod{13}$ $335 \times 608 \times 853 \equiv 7 \pmod{13}$
Revision Table: Remainder of Product by 13
| Concept | Description | Application to Problem |
|---|---|---|
| Modular Arithmetic | System of arithmetic for integers, where numbers "wrap around" upon reaching a certain value (the modulus). | Used to work with remainders. |
| Product Rule for Remainders | Remainder of $(a \times b) \div n$ is the same as Remainder of $((a \div n \text{ Remainder}) \times (b \div n \text{ Remainder})) \div n$. | Breaks down the problem of finding the remainder of a large product into smaller steps. |
| Individual Remainders | Finding $a \pmod n$, $b \pmod n$, etc. | Calculated for 335, 608, and 853 when divided by 13. |
| Product of Individual Remainders | Multiplying the remainders found in the previous step. | Calculated as $10 \times 10 \times 8 = 800$. |
| Final Remainder | Finding the remainder of the product of individual remainders when divided by the modulus (13). | Calculated as $800 \div 13$ Remainder, which is 7. |
Additional Information: Properties of Modular Arithmetic
Modular arithmetic is a fundamental concept in number theory and is very useful for problems involving remainders. Here are some basic properties:
- Addition Property: If $a \equiv r_1 \pmod n$ and $b \equiv r_2 \pmod n$, then $a + b \equiv r_1 + r_2 \pmod n$.
- Subtraction Property: If $a \equiv r_1 \pmod n$ and $b \equiv r_2 \pmod n$, then $a - b \equiv r_1 - r_2 \pmod n$. (Note: If $r_1 - r_2$ is negative, add the modulus $n$ to get a non-negative remainder).
- Multiplication Property: If $a \equiv r_1 \pmod n$ and $b \equiv r_2 \pmod n$, then $a \times b \equiv r_1 \times r_2 \pmod n$. (This is the property used in our solution).
- Exponentiation Property: If $a \equiv r \pmod n$, then $a^k \equiv r^k \pmod n$ for any non-negative integer $k$.
These properties allow us to perform arithmetic operations on the remainders instead of the original numbers, which simplifies calculations, especially with large numbers.
Important Questions from Divisibility and Remainder
If a five digit number 247xy is divisible by 3, 7 and 11, then what is the value of (2y - 8x)?
If the seven-digit number 94x29y6 is divisible by 72, then what is the value of (2x + 3y) for x ≠ y ?
Find the greatest value of b so that 30a68b (a > b) is divisible by 11.
- What is the least square number which is exactly divisible by 2, 3, 10, 18 and 20?
Find the greatest 3-digit number which, when divided by 3, 4, 5 and 8, leaves remainder 2 in each case.
