Exercise 1: Checking 158 ≡ 38 (mod 12) Two Ways — Possible Solution
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GIVEN
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a = 158, b = 38, n = 12

METHOD 1: SAME-REMAINDER DEFINITION
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158 mod 12 = 2 (since 158 = 12*13 + 2)
38 mod 12 = 2 (since 38 = 12*3 + 2)

Both leave remainder 2 when divided by 12 - the same remainder,
so by this definition, 158 ≡ 38 (mod 12) is TRUE.

METHOD 2: THE n | (a-b) DEFINITION
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a - b = 158 - 38 = 120

Check whether 12 divides 120: 120 / 12 = 10 exactly, with no
remainder. So 12 | 120, meaning this definition also confirms
158 ≡ 38 (mod 12) is TRUE.

RESULT
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Both definitions agree: 158 ≡ 38 (mod 12).

WHY THIS WORKS AS AN ANSWER
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Both of this chapter's own definitions of congruence are checked
independently rather than relying on just one, and both are shown
to produce the same true/false answer - directly confirming this
chapter's own claim that the two definitions are equivalent, not
just asserting it.