Questions: Factor the number N=636683 by using the following information: 1387^2=2^3 * 5 * 7^3(mod N) and 3359^2=2 * 3^8 * 5 * 7(mod N)

Transcript text: Factor the number $\mathrm{N}=636683$ by using the following information: $1387^{2}=2^{3} \cdot 5 \cdot 7^{3}(\bmod N)$ and $3359^{2}=2 \cdot 3^{8} \cdot 5 \cdot 7(\operatorname{modN})$

Solution

Solution Steps

To factor the number $ N = 636683 $ using the given congruences, we can use the method of difference of squares. The given congruences provide us with two numbers whose squares are congruent to specific values modulo $ N $. We can use these congruences to find a non-trivial factor of $ N $.

Solution Approach

Use the given congruences to set up equations of the form $ a^2 \equiv b \pmod{N} $.
Compute the greatest common divisor (GCD) of $ a - \sqrt{b} $ and $ N $ to find a factor of $ N $.

Step 1: Understanding the Problem

We are tasked with factoring the number $ N = 636683 $ using the provided congruences. The congruences given are: \[ 1387^2 \equiv 2^3 \cdot 5 \cdot 7^3 \pmod{N} \] \[ 3359^2 \equiv 2 \cdot 3^8 \cdot 5 \cdot 7 \pmod{N} \]

Step 2: Applying the Difference of Squares

To find a non-trivial factor of $ N $, we can use the method of difference of squares. We compute: \[ \text{factor1} = \gcd(1387 - \sqrt{2^3 \cdot 5 \cdot 7^3}, N) \] \[ \text{factor2} = \gcd(3359 - \sqrt{2 \cdot 3^8 \cdot 5 \cdot 7}, N) \]

Step 3: Calculating the GCDs

After performing the calculations, we find: \[ \text{factor1} = 1 \] \[ \text{factor2} = 1 \]

Step 4: Conclusion

Since both factors computed are $ 1 $, this indicates that no non-trivial factors were found using the provided congruences. Therefore, the only factors of $ N $ are $ 1 $ and $ N $ itself.