To find the simplest possible form of each number under the given moduli, we need to calculate the equivalent positive remainder for each negative number.
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For -17 (mod 2):
- First, we find the positive representative by adding the modulus until we get a non-negative number.
- -17 + 2 = -15
- -15 + 2 = -13
- -13 + 2 = -11
- -11 + 2 = -9
- -9 + 2 = -7
- -7 + 2 = -5
- -5 + 2 = -3
- -3 + 2 = -1
- -1 + 2 = 1
- Therefore, \(-17 \equiv 1 \mod 2\).
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For -56 (mod 13):
- We will add 13 until we get a non-negative number.
- -56 + 13 = -43
- -43 + 13 = -30
- -30 + 13 = -17
- -17 + 13 = -4
- -4 + 13 = 9
- Therefore, \(-56 \equiv 9 \mod 13\).
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For -50 (mod 4):
- Again, adding the modulus 4 until we get a non-negative number:
- -50 + 4 = -46
- -46 + 4 = -42
- -42 + 4 = -38
- -38 + 4 = -34
- -34 + 4 = -30
- -30 + 4 = -26
- -26 + 4 = -22
- -22 + 4 = -18
- -18 + 4 = -14
- -14 + 4 = -10
- -10 + 4 = -6
- -6 + 4 = -2
- -2 + 4 = 2
- Therefore, \(-50 \equiv 2 \mod 4\).
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For -75 (mod 7):
- Adding 7 until we reach a non-negative number:
- -75 + 7 = -68
- -68 + 7 = -61
- -61 + 7 = -54
- -54 + 7 = -47
- -47 + 7 = -40
- -40 + 7 = -33
- -33 + 7 = -26
- -26 + 7 = -19
- -19 + 7 = -12
- -12 + 7 = -5
- -5 + 7 = 2
- Therefore, \(-75 \equiv 2 \mod 7\).
In summary, the simplified forms are:
- \(-17 \equiv 1 \mod 2\)
- \(-56 \equiv 9 \mod 13\)
- \(-50 \equiv 2 \mod 4\)
- \(-75 \equiv 2 \mod 7\)