What is the remainder when the integer N = 3^{83} + 7^{83} is divided by 100? — GMAT

What is the remainder when the integer $N = 3^{83} + 7^{83}$ is divided by $100$ ?

Answer: 70

The remainder when

N = 3^{83} + 7^{83}

is divided by

100

is 70.

Finding the remainder when an integer is divided by

100

is equivalent to finding its last two digits. Powers of

3

repeat their last two digits in cycles of

20

, and

3^{20} \equiv 1 \pmod{100}

. Thus

3^{83} \equiv 3^3 = 27 \pmod{100}

. Powers of

7

repeat their last two digits in cycles of

4

, with

7^4 = 2401 \equiv 1 \pmod{100}

. Thus

7^{83} \equiv 7^3 = 343 \equiv 43 \pmod{100}

. Combining these gives

27 + 43 = 70

Find the remainder of

3^{83}

modulo

100

by identifying the cyclicity pattern of powers of

3

3^{20} \equiv 1 \pmod{100}

, which implies

3^{83} = (3^{20})^4 \cdot 3^3 \equiv 1^4 \cdot 27 = 27 \pmod{100}

Since

3^{20} \equiv 1 \pmod{100}

, exponent

83

can be reduced to

83 \pmod{20} = 3

Find the remainder of

7^{83}

modulo

100

by identifying the cyclicity pattern of powers of

7

7^4 = 2401 \equiv 1 \pmod{100}

, which implies

7^{83} = (7^4)^{20} \cdot 7^3 \equiv 1^{20} \cdot 343 \equiv 43 \pmod{100}

Since

7^4 \equiv 1 \pmod{100}

, exponent

83

can be reduced to

83 \pmod{4} = 3

Sum the modular remainders to obtain the final remainder of

N

N = 3^{83} + 7^{83} \equiv 27 + 43 = 70 \pmod{100}

The remainder of a sum is equal to the sum of the individual remainders modulo

100

Modular Arithmetic and Cyclicity of Last Two Digits

Question