An integer sequence is defined by T_n = 7^n - (-2)^n for all positive integers n. What is … — GMAT

An integer sequence is defined by $T_n = 7^n - (-2)^n$ for all positive integers $n$ . What is the remainder when $T_{40}$ is divided by $5$ ?

The remainder when

T_{40}

is divided by

5

0

Modulo

5

7 \equiv 2

, so

7^{40} \equiv 2^{40}

. Since

40

is an even exponent,

(-2)^{40} = 2^{40}

. Substituting these into the formula yields

T_{40} \equiv 2^{40} - 2^{40} = 0 \pmod 5

. Thus, the remainder is

0

Reduce the base

7

modulo

5

7 \equiv 2 \pmod 5

, so

7^{40} \equiv 2^{40} \pmod 5

Simplifying the base makes modular exponentiation straightforward.

Evaluate the negative base term

(-2)^{40}

Since

40

is an even integer,

(-2)^{40} = 2^{40}

An even power of a negative number yields a positive result.

Compute

T_{40}

modulo

5

T_{40} = 7^{40} - (-2)^{40} \equiv 2^{40} - 2^{40} = 0 \pmod 5

Subtracting identical values yields

0

Modular Arithmetic and Exponent Parity

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