Exercise 2: Checking e=19 and Computing d for phi(n)=46,620 — Possible Solution
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GIVEN
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phi(n) = 46620. Candidate e = 19.

STEP 1: CHECK gcd(19, 46620)
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19 is prime, and 46620 is not a multiple of 19 (46620 / 19 =
2453.68..., not a whole number) - so gcd(19, 46620) = 1.

e=19 is VALID.

STEP 2: RUN THE EXTENDED EUCLIDEAN ALGORITHM ON (19, 46620)
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q=0: r=46620, s=0    (initial swap step, since 19 < 46620)
q=2453: r=19, s=1
q=1: r=13, s=-2453
q=2: r=6, s=2454
q=6: r=1, s=-7361

The remainder reaches 1, confirming gcd=1 again directly from the
algorithm itself, matching the quick check in Step 1.

STEP 3: REDUCE TO GET d
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The coefficient for 19 is x = -7361.

d = -7361 mod 46620 = 39259

CHECK
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19 * 39259 = 745921
745921 mod 46620 = 1 (since 745921 = 46620*16 + 1)

Confirmed: e*d ≡ 1 (mod phi(n)).

RESULT
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e=19 is a valid public exponent, and its corresponding private
exponent is d=39259.

WHY THIS WORKS AS AN ANSWER
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The gcd check is performed both by quick inspection and confirmed
by the extended Euclidean algorithm's own final remainder,
providing two independent confirmations that e=19 is valid, and
the full step-by-step trace is shown rather than only the final d
value, matching this chapter's own established level of detail.