What is the unit digit of the numerical expression N = (137^{137} \times 264^{102}) + 619^… — State PSC Exam

What is the unit digit of the numerical expression $N = (137^{137} \times 264^{102}) + 619^{199}$ ?

Answer: 1

The unit digit of

137^{137}

is derived from

137 \pmod 4 = 1

, giving

7^1 = 7

. The unit digit of

264^{102}

is 6 because the exponent 102 is even. Multiplying these unit digits gives

7 \times 6 = 42

, which contributes a unit digit of 2. The unit digit of

619^{199}

is 9 because 199 is odd. Adding the unit digits yields

2 + 9 = 11

, whose unit digit is 1.

Determine the unit digit of

137^{137}

using the cyclicity of 7.

The exponent 137 leaves a remainder of 1 when divided by 4, giving a unit digit of

7^1 = 7

The unit digit pattern for powers of 7 repeats every 4 powers (7, 9, 3, 1).

Determine the unit digit of

264^{102}

using the cyclicity of 4.

Since 102 is an even exponent,

4^{\text{even}}

yields a unit digit of 6.

Powers of 4 end in 4 for odd exponents and 6 for even exponents.

Calculate the unit digit of the product

(137^{137} \times 264^{102})

(7 \times 6) = 42

, which has a unit digit of 2.

The unit digit of a product depends only on the product of the unit digits of its factors.

Determine the unit digit of

619^{199}

using the cyclicity of 9.

Since 199 is an odd exponent,

9^{\text{odd}}

yields a unit digit of 9.

Powers of 9 end in 9 for odd exponents and 1 for even exponents.

Sum the unit digits to find the final unit digit of expression

N

2 + 9 = 11

, which has a unit digit of 1.

The unit digit of the sum of two terms is the unit digit of the sum of their individual unit digits.

Unit digit determination using exponent cyclicity rules for base digits 4, 7, and 9.

Estimated Time:1m 15s

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