## GMAT Unit DIgit

Write Extended Concept

Course: GMAT

Topic: Unit Digit – Number Properties

## Unit Digit – Number Properties (Part 1)

The unit digit of a number refers to the digit in one place. In the context of GMAT questions, understanding the properties of unit digits is essential for solving various problems related to number properties, divisibility, and pattern recognition. In this part, we will discuss the basics of unit digits, their cyclical patterns, and how to apply this knowledge to solve GMAT problems.

**1. Basic properties of unit digits:**

The unit digit of a number does not depend on the other digits of the number. For example, the unit digit of 147 is 7, the unit digit of 2895 is 5, and the unit digit of 103 is 3. When performing operations such as addition, subtraction, multiplication, or exponentiation, we can focus solely on the unit digits to determine the unit digit of the result.

For addition and subtraction, the unit digit of the result can be found by performing the operation on the unit digits alone. For example:

- The unit digit of 17 + 23 is the same as the unit digit of 7 + 3, which is 0.
- The unit digit of 45 – 18 is the same as the unit digit of 5 – 8, which is 7 (if we borrow 1 from the tens digit, it becomes 15 – 8).

For multiplication, the unit digit of the product can be found by multiplying the unit digits of the two numbers. For example:

- The unit digit of 12 × 13 is the same as the unit digit of 2 × 3, which is 6.
- The unit digit of 27 × 49 is the same as the unit digit of 7 × 9, which is 3.

**2. Cyclical patterns of unit digits:**

When raising a number to a power, the unit digit of the result often follows a cyclical pattern. For example, consider the unit digits of the powers of 2:

- 2¹: unit digit 2
- 2²: unit digit 4
- 2³: unit digit 8
- 2⁴: unit digit 6
- 2⁵: unit digit 2
- 2⁶: unit digit 4
- and so on…

As you can see, the unit digits of the powers of 2 follow a cyclical pattern of 2-4-8-6. The cycle repeats every 4 powers. The same cyclical pattern applies to other numbers as well. Understanding these patterns can help you quickly identify the unit digit of a number raised to a power without having to compute the complete result.

In the next part, we will discuss more about the cyclical patterns of different numbers and how to use them to solve GMAT problems.

**Unit Digit – Number Properties (Part 2)**

In the first part, we introduced the concept of unit digits and their basic properties. We also briefly discussed cyclical patterns of unit digits when raising a number to a power. This part will delve deeper into the cyclical patterns for different numbers and show how to use these patterns to solve GMAT problems.

### 3. Cyclical patterns for different unit digits:

The unit digit of a number raised to a power follows a specific cyclical pattern. Here are the cyclical patterns for unit digits from 0 to 9:

- 0: always 0
- 1: always 1
- 2: 2-4-8-6 (repeats every 4 powers)
- 3: 3-9-7-1 (repeats every 4 powers)
- 4: 4-6 (repeats every 2 powers)
- 5: always 5
- 6: always 6
- 7: 7-9-3-1 (repeats every 4 powers)
- 8: 8-4-2-6 (repeats every 4 powers)
- 9: 9-1 (repeats every 2 powers)

Note that the cycles for 0, 1, 5, and 6 always result in the same unit digit, while the cycles for 2, 3, 7, and 8 repeat every 4 powers. The cycles for 4 and 9 repeat every 2 powers.

### 4. Using cyclical patterns to find the unit digit of a number raised to a power:

To determine the unit digit of a number raised to a power, follow these steps:

**Step 1:** Identify the unit digit of the base number.

**Step 2:** Determine the cycle of the unit digit.

**Step 3:** Find the remainder when the cycle length divides the exponent. Step 4: Use the remainder to identify the corresponding unit digit in the cycle.

##### Example:

**Find the unit digit of 3 ^{27}.**

**Step 1: **The unit digit of the base number is 3. Step 2: The cycle for 3 is 3-9-7-1 (repeats every 4 powers). Step 3: Divide the exponent (27) by the length of the cycle (4) to find the remainder: 27 ÷ 4 = 6 with a remainder of 3. Step 4: The remainder is 3, which corresponds to the third unit digit in the cycle: 7. Therefore, the unit digit of 3^{27} is 7.

By understanding cyclical patterns of unit digits, you can efficiently solve GMAT problems involving large numbers raised to power. In the next part, we will explore more complex GMAT problems that affect unit digits and learn how to tackle them.

**Unit Digit – Number Properties (Part 3)**

In the previous parts, we discussed the concept of unit digits, their basic properties, and the cyclical patterns for different numbers when raised to a power. In this part, we will explore more complex GMAT problems involving unit digits and learn how to tackle them using the concepts we’ve learned.

**Unit digit of the sum or difference of large numbers:**

In some GMAT problems, you might be asked to find the unit digit of the sum or difference of two or more significant numbers, possibly involving exponents. In such cases, you can use the properties of unit digits and cyclical patterns to simplify the calculations.

##### Example:

Find the unit digit of (5^{26} – 3^{33} + 7^{14}).

Solution:

First, find the unit digit of each term:

- For 5
^{26}, since the unit digit of 5 always remains 5, the unit digit is 5. - For 3
^{33}, the cycle for 3 is 3-9-7-1 (repeats every 4 powers). Divide the exponent (33) by the length of the cycle (4) to find the remainder: 33 ÷ 4 = 8 with a remainder of 1. The remainder is 1, which corresponds to the first unit digit in the cycle: 3. Therefore, the unit digit of 3^{33}is 3. - For 7
^{14}, the cycle for 7 is 7-9-3-1 (repeats every 4 powers). Divide the exponent (14) by the length of the cycle (4) to find the remainder: 14 ÷ 4 = 3 with a remainder of 2. The remainder is 2, which corresponds to the second unit digit in the cycle: 9. Therefore, the unit digit of 7^{14}is 9.

Now, find the unit digit of the sum or difference:

- (5
^{26}– 3^{33}+ 7^{14}) = (5 – 3 + 9), the unit digit is (5 – 3 + 9) = 11, so the unit digit is 1.

### The unit digit in a sequence or series:

In some problems, you might be asked to find the unit digit of a sequence or series, such as the sum of the first n terms of a sequence. In these cases, you can use the properties of unit digits and cyclical patterns to identify patterns in the sequence and then apply these patterns to find the unit digit of the sum or product.

**Example:**

Find the unit digit of the sum of the first 100 terms of the arithmetic sequence with the first term 3 and the common difference 7.

Solution:

First, find the unit digits of the sequence. Since the common difference is 7, the unit digits of the sequence will follow a cycle of 3-0-7-4-1-8-5-2-9-6 (repeats every 10 terms).

Next, notice that the sum of the unit digits in each cycle is 45. Since there are 10 cycles in the first 100 terms (100 ÷ 10 = 10), the unit digit of the sum of the first 100 terms is the same as the unit digit of 10 × 45, which is 0.

By understanding unit digit properties and cyclical patterns, you can efficiently solve a wide range of GMAT problems involving large numbers and sequences. In the final part, we will discuss some additional tips and strategies for solving unit digit problems on the GMAT.

## Unit Digit – Number Properties (Part 4)

In the previous parts, we discussed the concept of unit digits, their basic properties, cyclical patterns, and how to use them to solve various types of GMAT problems involving large numbers, exponents, sums, and sequences. In this final part, we will provide some additional tips and strategies for solving unit digit problems on the GMAT more effectively.

**Pay attention to the unit digit:**

When solving unit digit problems, focusing only on the unit digit and ignoring the rest of the number is essential. This helps you save time and avoid unnecessary calculations.

**Look for patterns:**

Unit digit problems often involve patterns, like the cyclical patterns we discussed earlier. Identifying these patterns early in the problem-solving process can make it much easier to find the correct solution.

**Practice mental calculations:**

Unit-digit problems often involve simple arithmetic operations on single-digit numbers. Practicing mental calculations can help you become more efficient and accurate when working with unit digits.

**Know the common cyclical patterns:**

Memorizing the common cyclical patterns for unit digits from 0 to 9 can help you quickly determine the unit digit of a number raised to a power. This knowledge can save you valuable time on the GMAT.

**Combine unit digit properties with other number properties:**

In some GMAT problems, you may need to combine unit digit properties with other number properties, such as divisibility rules or prime factorization. Being familiar with various number properties and knowing how to apply them in combination can help you solve more complex problems.

##### Example:

Find the unit digit of the product of the first 20 positive integers.

Solution:

The product of the first 20 positive integers is 20! (20 factorial). Notice that 20! includes the factors 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. Since each of these factors has a 2 in its prime factorization, the product of the first 20 positive integers will be divisible by 2^10. Additionally, 20! includes the factors 5, 10, 15, and 20. Since each of these factors has a 5 in its prime factorization, the product of the first 20 positive integers will be divisible by 5^4. Since the product is divisible by both 2^10 and 5^4, it must be divisible by 10^4 (10,000). Thus, the unit digit of the product of the first 20 positive integers is 0.

By mastering the properties of unit digits, cyclical patterns, and other number properties, you can efficiently and accurately solve a wide range of GMAT problems involving large numbers, exponents, sums, sequences, and more. With diligent practice, you’ll be well-prepared to tackle unit digit problems on the GMAT and improve your overall quantitative score.

## Question types: Multiple Choice (Five answer choices; one correct answer)

#### Difficulty level: Easy

**Question 1:**

What is the unit digit of 7^{41}?

A) 1 B) 3 C) 7 D) 9 E) 5

Answer: D) 7

**Question 2:**

What is the unit digit of the sum of 1234 + 567 + 1999 + 2345?

A) 3 B) 5 C) 7 D) 9 E) 1

Answer: B) 5

**Question 3:**

What is the unit digit of the product of 6 × 8 × 3 × 7?

A) 1 B) 2 C) 4 D) 6 E) 8

Answer: C) 4

**Difficulty level: Medium**

**Question ****4****:**

What is the unit digit of 3^{100} + 4^{100}?

A) 0 B) 1 C) 4 D) 7 E) 9

Answer: B) 1

**Question ****5****:**

What is the unit digit of the product of the first ten positive odd integers?

A) 1 B) 3 C) 5 D) 7 E) 9

Answer: E) 9

**Question ****6****:**

What is the unit digit of 5^{5} × 7^{7} × 2^{5} × 3^{3}?

A) 0 B) 2 C) 4 D) 6 E) 8

Answer: D) 6

**Difficulty level: Hard**

**Question ****7****:**

What is the unit digit of 2^{77} + 6^{88} – 3^{99}?

A) 0 B) 1 C) 2 D) 3 E) 9

Answer: C) 2

**Question ****8****:**

If the unit digit of the integer x is 7 and the unit digit of the integer y is 3, what is the unit digit of x^{3} + y^{3}?

A) 0 B) 1 C) 6 D) 8 E) 9

Answer: B) 1

**Question ****9****:**

If the unit digit of the integer a is 9 and the unit digit of the integer b is 2, what is the unit digit of the integer (a × b)^{3}?

A) 2 B) 3 C) 6 D) 8 E) 9

Answer: D) 8

#### Difficulty level: Challenging

**Question 1****0****:**

What is the unit digit of the product 6^{13} × 7^{37} × 9^{56} × 3^{72}?

A) 1 B) 3 C) 6 D) 7 E) 9

Answer: A) 1

**Question ****11****:**

If the unit digit of the integer k is 8, what is the unit digit of (k + 3)^{3}?

A) 1 B) 3 C) 4 D) 6 E) 7

Answer: E) 7

**Question ****12****:**

If a and b are positive integers, and the unit digits of a^{5} and b^{5} are both 2, what is the unit digit of a^{5} + b^{5}?

A) 0 B) 1 C) 2 D) 4 E) 5

Answer: D) 4

## Question types: Data Sufficiency (Five answer choices; one correct answer)

##### Difficulty level: Easy

**Question 1:**

What is the unit digit of the positive integer n?

(1) n is a multiple of 9.

(2) n^{3} has a unit digit of 9.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

**Question 2:**

What is the unit digit of the positive integer m?

(1) The unit digit of m^{4} is 6.

(2) The unit digit of 3m is 9.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

**Question 3:**

What is the unit digit of the positive integer x?

(1) The unit digit of x^{2} is 1.

(2) The unit digit of 5x is 5.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Question 4:

What is the unit digit of the positive integer y?

(1) The unit digit of y^{3} is 8.

(2) The unit digit of 4y is 2.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** D) EACH statement ALONE is sufficient.

##### Difficulty level: Medium

Question 5:

What is the unit digit of the positive integer p?

(1) The unit digit of p^{3} is 7.

(2) The unit digit of p^{4} is 1.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Question 6:

What is the unit digit of the positive integer q?

(1) The unit digit of 3q is 6.

(2) The unit digit of q^{2} + q is 5.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

#### Difficulty level: Hard

**Question ****7****:**

What is the unit digit of the positive integer r?

(1) The unit digit of r^{5} + r^{3} is 6.

(2) The unit digit of 7r is 5.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Question 8:

What is the unit digit of the positive integer s?

(1) The unit digit of s^3 – s^2 is 0.

(2) The unit digit of 6s is 6.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation: From statement (1), we know that the unit digit of s^3 – s^2 is 0. However, more than this statement is needed because we cannot uniquely determine the unit digit of s. From statement (2), we know that the unit digit of 6s is 6. There needs to be more than this statement because we cannot uniquely determine the unit digit of s. However, when we consider both statements together, we can deduce that s has a unit digit of 6, as 6 × 6 = 36, with a unit digit of 6, and 6^3 – 6^2 has a unit digit of 0.

#### Difficulty level: Challenging

**Question ****9****:**

What is the unit digit of the positive integer x?

(1) The sum of the unit digit of x^{2} and the unit digit of x^{3} is 0.

(2) The unit digit of (x + 2)^{2} is 6.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

**Question ****10****:**

What is the unit digit of the positive integer y?

(1) The unit digit of y^{3} + y^{2} is 2.

(2) The unit digit of (y^{2} + y) multiplied by the unit digit of (y + 1) is 0.

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

D) EACH statement ALONE is sufficient.

E) Statements (1) and (2) TOGETHER are NOT sufficient.

**Answer:** C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.