Exercise 2: The Reduce-Anytime Property on 456789 x 987654 mod 100 — Possible Solution
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GIVEN
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a = 456789, b = 987654, n = 100

METHOD 1: MULTIPLY FULLY, THEN REDUCE
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a * b = 456789 * 987654 = 451,112,232,606

451,112,232,606 mod 100 = 6
(the last two digits of the full product are "06")

METHOD 2: REDUCE EACH OPERAND FIRST, THEN MULTIPLY, THEN REDUCE
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a mod 100 = 456789 mod 100 = 89 (the last two digits of a)
b mod 100 = 987654 mod 100 = 54 (the last two digits of b)

89 * 54 = 4,806

4,806 mod 100 = 6

RESULT
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Both methods give exactly 6.

WHY THIS WORKS AS AN ANSWER
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Both computations are carried out fully and independently - the
full, much larger multiplication in Method 1, and the much smaller
multiplication of only the reduced operands in Method 2 - and both
land on the identical final answer, directly demonstrating this
chapter's own reduce-anytime property rather than just quoting it.
Method 2 also makes concrete why the property is useful: 89*54 is
a far smaller multiplication to perform than 456789*987654.