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Congruence (number theory)

We will deal with congruence from the field of number theory in the following sections. The characters for equal and identical are briefly discussed, because we need the identical afterwards. Suitable examples should clarify everything. This article is part of our Mathematics section.

Before we get into congruence, we need to briefly consider equal and identical. The following graphic shows this in brief:

We know this is the same from simple equations: 3 + 4 = 7. We need the identical in another place and we are now looking at exactly this.

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Congruence from number theory

We have just dealt with the identical sign. Where do you need this? Well, the identical sign can be found in congruence from the field of number theory. The definition looks like this: Two natural numbers are congruent modulo m if they leave the same remainder r when divided by the natural number m. And what does that mean in German? We divide the number before and the number after the identical sign by what is given after "mod". The rest must be the same in both cases. Let's look at two tasks to illustrate this.

Task 1 or example 1:

  • To see if the calculation is correct, we divide both 12 and 26 by 7.
  • 12: 7 = 1 remainder 5. Reason: The 7 goes completely into the 12 once, 5 are left over. 7 1 + 5 = 12
  • 26: 7 = 3 remainder 5. Reason: The 7 goes completely into the 25 three times, 5 remained. 7 3 + 5 = 26
  • In both cases the remainder is 5, everything is correct.

Task 2 or example 2:

  • To check whether the calculation is correct, we divide both 15 and 4 by 11. In both cases, the remainder must be the same for the problem / example to be correct.
  • 15: 11 = 1 remainder 4. Reason: The 11 goes completely into the 15, 4 are left over. 11 1 + 4 = 15
  • 4: 11 = 0 remainder 4. Reason: The 11 does not go into the 4 at all, 4 remain. 11 0 + 4 = 4.
  • In both cases the remainder is 4, everything is correct.

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