What is the unit digit of the composite exponential expression S = 717^{100} + 424^{101} -… — State PSC Exam

What is the unit digit of the composite exponential expression $S = 717^{100} + 424^{101} - 818^{100}$ ?

The unit digit of the expression is 9.

Evaluating each term using its base cyclicity gives unit digits of 1 for

717^{100}

, 4 for

424^{101}

, and 6 for

818^{100}

. Combining them yields

1 + 4 - 6 = -1

. Adding 10 to adjust for borrowing in unit digit subtraction gives

-1 + 10 = 9

Find the unit digit of

717^{100}

Unit digit is 1

The unit digit of the base is 7. The cyclicity of 7 is 4 (

7^1=7, 7^2=9, 7^3=3, 7^4=1

). Dividing the exponent 100 by 4 leaves remainder 0, which corresponds to the 4th power:

7^4 \equiv 1 \pmod{10}

Find the unit digit of

424^{101}

Unit digit is 4

The unit digit of the base is 4. The cyclicity of 4 is 2 (

4^{\text{odd}}=4, 4^{\text{even}}=6

). Since the exponent 101 is odd, the unit digit is 4.

Find the unit digit of

818^{100}

Unit digit is 6

The unit digit of the base is 8. The cyclicity of 8 is 4 (

8^1=8, 8^2=4, 8^3=2, 8^4=6

). Dividing the exponent 100 by 4 leaves remainder 0, which corresponds to the 4th power:

8^4 \equiv 6 \pmod{10}

Combine the unit digits to find the final result

Unit digit is 9

Substitute the individual unit digits into the expression:

1 + 4 - 6 = 5 - 6 = -1

. Converting

-1

to a positive unit digit modulo 10 gives

-1 + 10 = 9

Unit digit determination using cyclicity and modular arithmetic for composite exponential expressions.

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