Lesson 5, Topic 1
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# Modular Equivalence

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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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