Consider the numerical expression E = 8^{52} - 3^{27}. If u represents the unit digit of t… — State PSC Exam

Consider the numerical expression $E = 8^{52} - 3^{27}$ . If $u$ represents the unit digit of the positive integer $E$ , which of the following correctly classifies the number $u$ ?

It is an odd composite number.Cevap
B
It is an even composite number.
C
It is a negative integer.
D
It is a unique number that is neither prime nor composite.

Cevap

The calculated unit digit is 9, which is classified as an odd composite number.

Evaluating the expression requires finding the unit digits of both exponential terms. The unit digit of

8^{52}

is 6, and the unit digit of

3^{27}

is 7. When subtracting 7 from 6 in a larger positive number, borrowing from the tens place results in

16 - 7 = 9

. The number 9 is odd and has three distinct factors (1, 3, 9), making it an odd composite number.

Adım Adım Çözüm

Determine the unit digit of

8^{52}

The unit digit is 6.

The cyclicity pattern for powers of 8 is 4 (ending in 8, 4, 2, 6). Since 52 is a perfect multiple of 4 (remainder is 0), the unit digit matches the 4th power in the cycle, which is 6.

Determine the unit digit of

3^{27}

The unit digit is 7.

The cyclicity pattern for powers of 3 is 4 (ending in 3, 9, 7, 1). Dividing the exponent 27 by 4 leaves a remainder of 3. Therefore, the unit digit matches the 3rd power in the cycle (

3^3

), which ends in 7.

Calculate the unit digit

u

of the difference

E

u = 9

Subtracting the unit digits gives

6 - 7

. Because

E

is a positive integer, we must borrow 10 from the next higher place value in the base-10 system, yielding

16 - 7 = 9

Classify the resulting number 9.

9 is an odd composite number.

The integer 9 cannot be evenly divided by 2 (making it odd) and has positive divisors other than 1 and itself (1, 3, and 9), which classifies it as composite.

Anahtar Kavram

Integration of power cyclicity rules and fundamental number classification.

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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