- #1
ninjagod123
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Show that if a [tex]\equiv[/tex] b mod p for all primes p, then a = b.
JSuarez said:Well, a - b must be divisible by all primes p. What is the only way for this to happen?
ninjagod123 said:I guess a - b can't be the product of all primes?
A congruence is a mathematical concept that describes the relationship between two numbers or shapes that have the same size, shape, and orientation. In other words, two objects are congruent if they are exactly the same, but may be in different positions or orientations.
While equality means that two quantities are exactly the same, congruence allows for differences in position or orientation. For example, two triangles can be congruent if they have the same sides and angles, but they can be in different positions on a plane.
The process for solving congruences depends on the type of congruence. For proving a=b, the first step is to use the given information to identify any known congruent sides or angles. Then, use properties of congruent triangles, such as the Side-Side-Side (SSS) or Angle-Side-Angle (ASA) congruence criteria, to prove that the remaining sides and angles are congruent.
To check if your solution to a congruence is correct, you can use substitution. Simply plug in the values for the congruent sides and angles into the original congruence statement. If the values are equal, then your solution is correct.
Yes, there can be more than one solution to a congruence. This is because there may be multiple ways to prove that two objects are congruent. However, it is important to use all given information and follow the appropriate congruence criteria to ensure that your solution is correct.