Discussion

Explanation:

Here, 6 and 14 have a common factor i.e., 2.

⇒ R[6¹⁰⁰/14] = R[2¹⁰⁰×3¹⁰⁰/14] = 2 × R[2⁹⁹×3¹⁰⁰/7] = 2 × R[2⁹⁹/7] × R[3¹⁰⁰/7]

Now, R[3¹⁰⁰/7] = 4

and, R[2⁹⁹/7] = 1

∴ R[6¹⁰⁰/14] = 2 × R[2⁹⁹/7] × R[3¹⁰⁰/7] = 2 × 1 × 4 = 8.

Alternately,

Let us find the pattern that remainders follow when successive powers of 6 are divided by 14.

Remainder when 6¹/14 = 6.
Remainder when 6²/14 = 8.
Remainder when 6³/14 = 6.
Remainder when 6⁴/14 = 8.

∴ We find that the remainders are repeated after every two powers.

⇒ R[6¹⁰⁰/14] = R[6²/14] = 8

Hence, 8.

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