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SS1: MATHEMATICS - 1ST TERM

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  1. Number Base System I | Week 1
    6Topics
    |
    2 Quizzes
  2. Number Base System II | Week 2
    3Topics
    |
    1 Quiz
  3. Number Base System III | Week 3
    2Topics
    |
    1 Quiz
  4. Modular Arithmetic I | Week 4
    2Topics
  5. Modular Arithmetic II | Week 5
    2Topics
  6. Modular Arithmetic III | Week 6
    3Topics
    |
    1 Quiz
  7. Indices I | Week 7
    2Topics
  8. Indices II | Week 8
    1Topic
    |
    1 Quiz
  9. Logarithms I | Week 9
    3Topics
  10. Logarithms II
    1Topic
    |
    1 Quiz
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Definition

Two integers are said to be modular equivalent if they have the same remainder in modular arithmetic.

Example 1

Show that the following pairs of numbers are equivalent in the moduli given: 

(i) 67 and 94 (mod9) (ii) 108 and 97 (mod11)

Solution

(i)

67 = 7 x 9 + 4

94 = 10 x 9 + 4

Note that the remainder is 4 in each case

Thus  \( \scriptsize 94 \equiv 67 ( mod 9) \)

The integers 67 and 94 are congruent because they have the same modular equivalent

(ii)

\( \scriptsize 108 \equiv 9 \: \times \: 11 \: + \: 9 \)

\(\scriptsize 97 \equiv 8 \: \times \: 11 \: + \: 9 \)

Thus  \( \scriptsize 97 \equiv 108 ( mod 11) \)

Hint:

In general, two integers x and y  are said to be congruent modulo z, i.e. 

\( \scriptsize x \equiv y ( mod z) \)

i.e. If the difference of x – y is an integer multiple of z, where the number z is referred to as the modulus of the congruence.

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