What is the unit digit of the composite expression E = (1! + 2! + 3! + \dots + 20!)^{2026}… — State PSC Exam

What is the unit digit of the composite expression $E = (1! + 2! + 3! + \dots + 20!)^{2026} + (3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20})$ ?

Answer: 8

Answer

The unit digit of the expression is 8.

Evaluating the expression requires breaking it down into two components. First, for the factorial sum

1! + 2! + 3! + \dots + 20!

, every term from

5!

onward contains factors of both 2 and 5, so its unit digit is 0. The unit digit of the sum is determined solely by

1! + 2! + 3! + 4! = 33

, which has a unit digit of 3. Raising 3 to the power 2026 gives

3^{2026}

. Since the unit digits of powers of 3 repeat in cycles of 4 (3, 9, 7, 1) and

2026 \equiv 2 \pmod 4

, the unit digit of

3^{2026}

3^2 = 9

.

Second, the product

3^1 \times 3^2 \times \dots \times 3^{20}

simplifies using the exponent addition rule to

3^{1+2+\dots+20} = 3^{210}

. Dividing 210 by 4 leaves a remainder of 2, so

3^{210}

also has a unit digit of

3^2 = 9

.

Adding the unit digits of both terms gives

9 + 9 = 18

, resulting in a final unit digit of 8.

Step-by-Step Solution

Find the unit digit of the inner factorial sum

S = 1! + 2! + 3! + \dots + 20!

The unit digit of

S

is 3.

For all

k \ge 5

k!

is divisible by 10 and ends in 0. Thus, only the sum of the first four terms

1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33

determines the unit digit.

Calculate the unit digit of the first term

S^{2026} \equiv 3^{2026}

The unit digit of the first term is 9.

The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent 2026 by 4 gives a remainder of 2 (

2026 = 4 \times 506 + 2

). Therefore, the unit digit is

3^2 = 9

Simplify the exponential product

P = 3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20}

The product simplifies to

3^{210}

By exponent multiplication rules,

3^1 \times 3^2 \times \dots \times 3^{20} = 3^{\sum_{i=1}^{20} i}

. The sum of the first 20 positive integers is

\frac{20 \times 21}{2} = 210

Calculate the unit digit of

3^{210}

The unit digit of the second term is 9.

Dividing the exponent 210 by 4 gives a remainder of 2 (

210 = 4 \times 52 + 2

). Therefore, the unit digit is

3^2 = 9

Combine the unit digits of the two terms.

The unit digit of

E

is 8.

Adding the unit digits gives

9 + 9 = 18

. The unit digit of 18 is 8.

Key Concept

Unit Digit and Cyclicity of Factorial and Exponential Expressions

Estimated Time:2m 0s

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