Find an integer having the remainders ## 2, 3, 4, 5 ##.

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In summary, the conversation discusses how an integer, represented by x, has four different congruences with different moduli. By finding the least common multiple of the moduli, which is 60, it is determined that the integer is congruent to 59 mod 60.
  • #1
Math100
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Homework Statement
Find an integer having the remainders ## 2, 3, 4, 5 ## when divided by ## 3, 4, 5, 6 ##, respectively. (Bhaskara, born ## 1114 ## ).
Relevant Equations
None.
Let ## x ## be an integer.
Then ## x\equiv 2\pmod {3}, x\equiv 3\pmod {4}, x\equiv 4\pmod {5} ## and ## x\equiv 5\pmod {6} ##.
This means
\begin{align*}
&x\equiv 2\pmod {3}\implies x+1\equiv 3\pmod {3}\implies x+1\equiv 0\pmod {3},\\
&x\equiv 3\pmod {4}\implies x+1\equiv 4\pmod {4}\implies x+1\equiv 0\pmod {4},\\
&x\equiv 4\pmod {5}\implies x+1\equiv 5\pmod {5}\implies x+1\equiv 0\pmod {5},\\
&x\equiv 5\pmod {6}\implies x+1\equiv 6\pmod {6}\implies x+1\equiv 0\pmod {6}.\\
\end{align*}
Observe that ## lcm(3, 4, 5, 6)=60 ##.
Thus ## x+1\equiv 0\pmod {60}\implies x\equiv -1\pmod {60}\implies x\equiv 59\pmod {60} ##.
Therefore, the integer is ## 59 ##.
 
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  • #2
Correct. It looks somehow familiar. Have we had ##59## as a solution before?
 
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fresh_42 said:
Correct. It looks somehow familiar. Have we had ##59## as a solution before?
I don't remember. It does not look familiar to me, though. Interesting to know that.
 

FAQ: Find an integer having the remainders ## 2, 3, 4, 5 ##.

1. What is the definition of remainders in mathematics?

In mathematics, remainders refer to the amount left over after dividing one number by another. For example, when dividing 10 by 3, the remainder is 1 because 3 goes into 10 three times with a remainder of 1.

2. How do you find an integer with specific remainders?

To find an integer with specific remainders, you can use the Chinese Remainder Theorem. This theorem states that if you have a system of linear congruences (equations with remainders), you can find a unique solution by using the remainders as coefficients and finding the least common multiple of the divisors.

3. Can you give an example of finding an integer with remainders 2, 3, 4, and 5?

Yes, for example, if you want to find an integer with remainders 2, 3, 4, and 5 when divided by 3, 4, 5, and 6 respectively, you can use the Chinese Remainder Theorem to find that the integer is 59. This means that when 59 is divided by 3, the remainder is 2, when divided by 4, the remainder is 3, and so on.

4. Are there any other methods for finding an integer with specific remainders?

Yes, there are other methods such as using modular arithmetic or the Euclidean algorithm. However, the Chinese Remainder Theorem is often the most efficient and simplest method to use.

5. Can you use the Chinese Remainder Theorem to find an integer with more than four remainders?

Yes, the Chinese Remainder Theorem can be used to find an integer with any number of remainders. The only requirement is that the divisors (numbers that the integer is being divided by) must be pairwise coprime, meaning they have no common factors.

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