Determine the unit digit of the composite exponential expression E = 17^{18^{19}} + 18^{19… — State PSC Exam

Determine the unit digit of the composite exponential expression $E = 17^{18^{19}} + 18^{19^{20}} - 19^{20^{21}}$ .

Answer: 8

The unit digit of

17^{18^{19}}

is 1 because

18^{19}

is divisible by 4, which corresponds to the 4th power

7^4 \equiv 1 \pmod{10}

. The unit digit of

18^{19^{20}}

is 8 because

19^{20} \equiv 1 \pmod 4

, giving

8^1 = 8

. The unit digit of

19^{20^{21}}

is 1 because

20^{21}

is an even exponent, giving

9^{\text{even}} \equiv 1 \pmod{10}

. Combining these gives

1 + 8 - 1 = 8

Find the unit digit of

17^{18^{19}}

The unit digit is 1

Base 7 has cyclicity 4. Exponent

18^{19} = 2^{19} \equiv 0 \pmod 4

. When the remainder modulo 4 is 0, we take the 4th power in the cycle:

7^4 \equiv 1 \pmod{10}

Find the unit digit of

18^{19^{20}}

The unit digit is 8

Base 8 has cyclicity 4. Exponent

19^{20} \equiv (-1)^{20} \equiv 1 \pmod 4

. The 1st power in the cycle gives

8^1 = 8

Find the unit digit of

19^{20^{21}}

The unit digit is 1

Base 9 has cyclicity 2. The exponent

20^{21}

is even, so

9^{\text{even}} \equiv 1 \pmod{10}

Evaluate the combined expression

The final unit digit is 8

Combine the unit digits of the individual terms:

1 + 8 - 1 = 8

Cyclicity of numbers and modular arithmetic for tower exponents

Question