Find the expansion base 7 of 67 and Compute (37^15 + 27 × 43) mod 6 using modular arithmetic.

Question

Find the expansion base 7 of 67

Compute (37¹⁵ + 27 × 43) mod 6 using modular arithmetic (without the use of a calculator). Show all your work.

Answer 1

The expansion of 67 in base 7 is as follows:

First, we note that 7² = 49, and then we subtract that from 67 to get the following equation:

67 (in base 10) = 49 + 18 = 7² + 14 + 4 = 1 * 7² + 2 * 7 + 4 * 1 = 124 (in base 7).

Now, we also have that: (37¹⁵ + 27 × 43) (mod 6) = 37¹⁵ (mod 6) + (27*43) (mod 6)

= [ 37 (mod 6) ]¹⁵ + 27 * [ 43 (mod 6) ] = [ 1 (mod 6) ]¹⁵ + 27 * [ 1 (mod 6) ]

= 1 (mod 6) + 27 (mod 6) = 28 (mod 6) = 4 (mod 6).