What is the unit digit of the numerical expression M = 3^{105} \times 7^{82} + 8^{96}? — State PSC Exam

What is the unit digit of the numerical expression $M = 3^{105} \times 7^{82} + 8^{96}$ ?

Answer: 3

The unit digit of the given expression is 3.

The unit digit of an exponential term is determined by its base's cyclicity cycle and the exponent modulo 4. For

3^{105}

105 \pmod 4 = 1 \implies 3^1 = 3

. For

7^{82}

82 \pmod 4 = 2 \implies 7^2 = 49 \implies 9

. The product of these terms has a unit digit of

(3 \times 9) \pmod{10} = 7

. For

8^{96}

96 \pmod 4 = 0

, which indicates the 4th position in the cyclicity pattern of 8 (8, 4, 2, 6), yielding a unit digit of 6. Finally, adding the unit digits gives

(7 + 6) \pmod{10} = 3

Determine the cyclicity and remainder for

3^{105}

Unit digit is 3

The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent by 4 gives

105 \pmod 4 = 1

, corresponding to

3^1 = 3

Determine the cyclicity and remainder for

7^{82}

Unit digit is 9

The unit digits of powers of 7 repeat in a cycle of 4 (7, 9, 3, 1). Dividing the exponent by 4 gives

82 \pmod 4 = 2

, corresponding to

7^2 = 49

(unit digit 9).

Calculate the unit digit of the product

3^{105} \times 7^{82}

Unit digit is 7

Multiply the unit digits of the two factors:

(3 \times 9) = 27

, which has a unit digit of 7.

Determine the cyclicity and remainder for

8^{96}

Unit digit is 6

The unit digits of powers of 8 repeat in a cycle of 4 (8, 4, 2, 6). The exponent 96 is completely divisible by 4 (

96 \pmod 4 = 0

). When remainder is 0, we use the 4th power in the cycle (

8^4 = 4096

), which ends in 6.

Sum the unit digits of the two main terms

Unit digit is 3

Adding the unit digits gives

(7 + 6) = 13

, which has a unit digit of 3.

Determining the unit digit of composite exponential expressions using power cyclicity.

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