Basic Numeracy Practice Questions — State PSC Exam — Page 12

Question 221Question →

What is the unit digit of the composite expression $E = (648^{116} \times 437^{85}) - 322^{90}$ ?

8

3

2

7

Show answer & explanation

Answer: 8

Answer

The unit digit of the expression is 8.

The term

648^{116}

has a unit digit of 6 (since

116 \pmod 4 = 0

, giving

8^4 \equiv 6 \pmod{10}

), and

437^{85}

has a unit digit of 7 (since

85 \pmod 4 = 1

, giving

7^1 \equiv 7 \pmod{10}

). The product of these two terms yields a unit digit of

(6 \times 7) \pmod{10} = 2

. The subtracted term

322^{90}

has a unit digit of 4 (since

90 \pmod 4 = 2

, giving

2^2 \equiv 4 \pmod{10}

). Subtracting gives

2 - 4 = -2 \equiv 8 \pmod{10}

.

Step-by-Step Solution

1

Find the unit digit of

648^{116}

Unit digit is 6

The base unit digit is 8, which follows a cyclicity of 4 (8, 4, 2, 6). Exponent

116 \div 4 = 29

with remainder 0. A remainder of 0 corresponds to the 4th position in the cyclicity cycle (

8^4 = 4096

), giving a unit digit of 6.

2

Find the unit digit of

437^{85}

Unit digit is 7

The base unit digit is 7, which follows a cyclicity of 4 (7, 9, 3, 1). Exponent

85 \div 4 = 21

with remainder 1. A remainder of 1 corresponds to the 1st power (

7^1 = 7

), giving a unit digit of 7.

3

Calculate the unit digit of the product

(648^{116} \times 437^{85})

Unit digit is 2

Multiplying the unit digits from Step 1 and Step 2 gives

6 \times 7 = 42

, which has a unit digit of 2.

4

Find the unit digit of

322^{90}

Unit digit is 4

The base unit digit is 2, which follows a cyclicity of 4 (2, 4, 8, 6). Exponent

90 \div 4 = 22

with remainder 2. A remainder of 2 corresponds to the 2nd power (

2^2 = 4

), giving a unit digit of 4.

5

Subtract to find the overall unit digit of

E

Unit digit is 8

Subtracting the unit digits gives

2 - 4 = -2

. Converting to a positive unit digit in base 10 arithmetic gives

-2 + 10 = 8

.

Key Concept

Unit Digit and Cyclicity Rules for Exponential Expressions

Question 222Question →

What is the simplified value of the following mathematical expression?

45 \div 5 \times 3 + [18 - \{14 - (8 - 3)\}]

Show answer & explanation

Answer: 36

Answer

36

Following the BODMAS rule step by step:
1. Innermost parenthesis:

8 - 3 = 5

.
2. Curly braces:

14 - 5 = 9

.
3. Square brackets:

18 - 9 = 9

.
4. Left-to-right division and multiplication:

45 \div 5 \times 3 = 9 \times 3 = 27

.
5. Addition:

27 + 9 = 36

.

Step-by-Step Solution

1

Evaluate the innermost round brackets

8 - 3 = 5

According to the BODMAS rule, operations inside the innermost brackets must be solved first.

2

Evaluate the expression inside the curly braces

14 - 5 = 9

Substitute the result from the round brackets into the curly braces.

3

Evaluate the expression inside the square brackets

18 - 9 = 9

Substitute the result from the curly braces into the square brackets.

4

Perform Division and Multiplication from left to right

45 / 5 * 3 = 9 * 3 = 27

Division and multiplication have equal precedence and must be evaluated strictly from left to right.

5

Add the result of the bracket terms to the result of the arithmetic terms

27 + 9 = 36

Final addition yields the simplified result.

Key Concept

BODMAS Rule (Brackets, Orders, Division and Multiplication left-to-right, Addition and Subtraction left-to-right)

Question 223Question →

What is the value of the following mathematical expression?

25 - [18 - \{12 - (6 - \overline{4 - 1})\}]

16

18

22

12

Show answer & explanation

Answer: 16

Answer

16

Following the BODMAS priority (Vinculum

\rightarrow

Round Brackets

\rightarrow

Curly Braces

\rightarrow

Square Brackets

\rightarrow

Outermost Subtraction):
1. Vinculum:

\overline{4 - 1} = 3

2. Round brackets:

6 - 3 = 3

3. Curly braces:

12 - 3 = 9

4. Square brackets:

18 - 9 = 9

5. Outer subtraction:

25 - 9 = 16

.
Thus, 16 is the mathematically correct answer.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\overline{4 - 1} = 3

According to BODMAS, vinculum (bar bracket) has the highest priority.

2

Simplify the innermost round brackets

(6 - 3)

6 - 3 = 3

Evaluate terms inside round brackets next.

3

Simplify the curly braces

\{12 - 3\}

12 - 3 = 9

Evaluate terms inside curly braces next.

4

Simplify the square brackets

[18 - 9]

18 - 9 = 9

Evaluate terms inside square brackets next.

5

Perform final subtraction from 25

25 - 9 = 16

Complete the outermost operation.

Key Concept

BODMAS Rule with Vinculum (Bar Bracket)

Question 224Question →

What is the unit digit of the expression $2^{44}$ ?

1

2

6

8

Show answer & explanation

Answer: 6

Answer

The unit digit of

2^{44}

is 6.

The unit digit of powers of 2 repeats in a pattern of 4 steps: 2, 4, 8, 6. Dividing the exponent 44 by 4 yields a remainder of 0. A remainder of 0 means the full 4th power in the cycle applies, so

2^4 = 16

yields a unit digit of 6.

Step-by-Step Solution

1

Identify the base and its cyclicity pattern.

The base is 2. The powers of 2 follow a cyclic pattern of unit digits:

2^1 = 2

,

2^2 = 4

,

2^3 = 8

,

2^4 = 16

(unit digit 6). The cyclicity of 2 is 4.

Unit digits repeat in a fixed pattern of length 4 for base 2.

2

Divide the exponent by the cyclicity period.

44 \div 4 = 11

with a remainder of

0

.

Determining the remainder tells us which position in the 4-step cycle applies.

3

Apply the cyclicity rule for remainder 0.

A remainder of 0 corresponds to the 4th power in the cycle (

2^4 = 16

), giving a unit digit of 6.

When the exponent is exactly divisible by the cyclicity length, the unit digit is determined by the last element of the cycle (

2^4

), not

2^0

.

Key Concept

Unit Digit Cyclicity for Base 2

Question 225Question →

What is the simplified value of the following mathematical expression?

\frac{7}{2} \div \left[ 1 \frac{1}{4} + \left\{ 2 \frac{1}{2} - \frac{1}{2} \times \left( 2.5 - \overline{\frac{1}{4} + \frac{1}{6}} \right) \right\} \right] \times \frac{5}{12}

\frac{7}{13}

\frac{35}{61}

\frac{1008}{325}

\frac{5}{13}

Show answer & explanation

Answer: $\frac{7}{13}$

Answer

The simplified value of the expression is

\frac{7}{13}

.

Following strict BODMAS precedence: First, simplify the bar term

\overline{\frac{1}{4} + \frac{1}{6}} = \frac{5}{12}

. Next, resolve the round bracket

\frac{5}{2} - \frac{5}{12} = \frac{25}{12}

. Multiplying by

\frac{1}{2}

gives

\frac{25}{24}

. The curly bracket simplifies to

\frac{5}{2} - \frac{25}{24} = \frac{35}{24}

. The square bracket yields

\frac{5}{4} + \frac{35}{24} = \frac{65}{24}

. Finally, evaluating left to right gives

\frac{7}{2} \times \frac{24}{65} \times \frac{5}{12} = \frac{7}{13}

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar bracket)

\overline{\frac{1}{4} + \frac{1}{6}} = \frac{3 + 2}{12} = \frac{5}{12}

According to BODMAS, the vinculum takes precedence over round brackets.

2

Evaluate the terms inside the round brackets

2.5 - \frac{5}{12} = \frac{5}{2} - \frac{5}{12} = \frac{30 - 5}{12} = \frac{25}{12}

Convert decimal

2.5

to fraction

\frac{5}{2}

and perform subtraction.

3

Perform multiplication inside the curly brackets

\frac{1}{2} \times \frac{25}{12} = \frac{25}{24}

Multiplication inside brackets precedes subtraction.

4

Evaluate the terms inside the curly brackets

2 \frac{1}{2} - \frac{25}{24} = \frac{5}{2} - \frac{25}{24} = \frac{60 - 25}{24} = \frac{35}{24}

Complete the subtraction inside the curly brackets.

5

Evaluate the terms inside the square brackets

1 \frac{1}{4} + \frac{35}{24} = \frac{5}{4} + \frac{35}{24} = \frac{30 + 35}{24} = \frac{65}{24}

Perform addition within the square brackets.

6

Perform division and multiplication from left to right

\frac{7}{2} \div \frac{65}{24} \times \frac{5}{12} = \frac{7}{2} \times \frac{24}{65} \times \frac{5}{12} = \frac{84}{65} \times \frac{5}{12} = \frac{7}{13}

Division and multiplication have equal precedence and are executed strictly from left to right.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets

Question 226Question →

Let $N$ be the smallest positive integer that leaves remainders of $2$ , $4$ , and $6$ when divided by $5$ , $7$ , and $9$ , respectively. What is the remainder when $N^{2025}$ is divided by $17$ ?

11

6

2

5

Show answer & explanation

Answer: 11

Answer

The remainder when

N^{2025}

is divided by

17

is 11.

The number

N

satisfies

N \equiv -3 \pmod{\text{lcm}(5,7,9)}

, giving

N = 315 - 3 = 312

. Reducing

312 \pmod{17}

gives

6

. By Fermat's Little Theorem,

6^{16} \equiv 1 \pmod{17}

, so

6^{2025} \equiv 6^9 \pmod{17}

. Calculating

6^9 \pmod{17}

yields

-6 \pmod{17}

, which converts to positive remainder

17 - 6 = 11

.

Step-by-Step Solution

1

Determine the value of the smallest positive integer

N

N = 312

From the problem statement:

N \equiv 2 \equiv -3 \pmod 5

,

N \equiv 4 \equiv -3 \pmod 7

, and

N \equiv 6 \equiv -3 \pmod 9

. Therefore,

N + 3

must be divisible by

\text{lcm}(5, 7, 9) = 315

. The smallest positive integer is

N = 315 - 3 = 312

.

2

Reduce

N

modulo

17

N \equiv 6 \pmod{17}

Dividing

312

by

17

gives

312 = 17 \times 18 + 6

, so

312 \equiv 6 \pmod{17}

.

3

Apply Fermat's Little Theorem to reduce the power

6^{2025} \equiv 6^9 \pmod{17}

Since

17

is prime and

\gcd(6, 17) = 1

,

6^{16} \equiv 1 \pmod{17}

. Expressing the exponent as

2025 = 16 \times 126 + 9

yields

6^{2025} \equiv (6^{16})^{126} \times 6^9 \equiv 1^{126} \times 6^9 \equiv 6^9 \pmod{17}

.

4

Evaluate

6^9 \pmod{17}

and convert to positive remainder

Remainder is

11

Computing successive powers modulo

17

:

6^2 = 36 \equiv 2 \pmod{17}

,

6^4 \equiv 2^2 = 4 \pmod{17}

, and

6^8 \equiv 4^2 = 16 \equiv -1 \pmod{17}

. Thus,

6^9 = 6^8 \times 6 \equiv (-1) \times 6 = -6 \pmod{17}

. Converting to a positive remainder gives

-6 + 17 = 11

.

Key Concept

Chinese Remainder Theorem (Constant Difference Method), Fermat's Little Theorem, and Negative Remainder Conversion

Question 227Question →

What is the unit digit of the expression $N = 2^{40} + 4^{31} - 7^{38}$ ?

1

6

9

3

Show answer & explanation

Answer: 1

Answer

The unit digit of the given expression is 1.

The unit digits of the individual terms are 6, 4, and 9 respectively. Adding the first two unit digits gives

6 + 4 = 10

(unit digit 0). Subtracting 9 from 10 yields

10 - 9 = 1

. Therefore, the overall unit digit of the expression is 1.

Step-by-Step Solution

1

Find the unit digit of

2^{40}

using cyclicity.

The cyclicity of 2 is 4 (2, 4, 8, 6). Divide the exponent 40 by 4:

40 \pmod 4 = 0

. When the remainder is 0, take the 4th power:

2^4 = 16

, so the unit digit is 6.

Powers of 2 repeat their unit digits in cycles of 4.

2

Find the unit digit of

4^{31}

.

The cyclicity of 4 is 2 (4 for odd exponents, 6 for even exponents). Since 31 is odd, the unit digit is 4.

Odd powers of 4 always end in 4.

3

Find the unit digit of

7^{38}

.

The cyclicity of 7 is 4 (7, 9, 3, 1). Divide the exponent 38 by 4:

38 \pmod 4 = 2

.

7^2 = 49

, so the unit digit is 9.

A remainder of 2 corresponds to the second term in the cyclicity sequence of 7.

4

Combine the unit digits according to the expression

N = 2^{40} + 4^{31} - 7^{38}

.

Unit digit =

(6 + 4) - 9 = 10 - 9 = 1

.

Perform modular arithmetic modulo 10 to find the final unit digit.

Key Concept

Unit Digit Cyclicity and Modular Addition/Subtraction

Question 228Question →

What is the unit digit of $7^{82}$ ?

Show answer & explanation

Answer: 9

Answer

The unit digit of

7^{82}

is 9.

The unit digits of powers of 7 repeat in a pattern of 4 (7, 9, 3, 1). Dividing the exponent 82 by 4 yields a remainder of 2. The second number in the cyclic pattern is 9, making 9 the unit digit of

7^{82}

.

Step-by-Step Solution

1

Find the cyclicity of the base number 7

The unit digits follow a repeating 4-step sequence: 7, 9, 3, 1.

Powers of 7 cycle every 4 powers because

7^1=7

,

7^2=49

,

7^3=343

, and

7^4=2401

.

2

Divide the exponent by the cyclicity period

82 \div 4 = 20

with a remainder of 2.

The remainder determines the equivalent power position within the 4-step cycle.

3

Determine the unit digit from the remainder

The unit digit of

7^2

is 9.

A remainder of 2 corresponds to

7^2

, giving 9.

Key Concept

Unit Digit Cyclicity

Question 229Question →

Let $N = 75x38y2$ be a seven-digit number that is completely divisible by $72$ , where $x$ and $y$ are single-digit natural numbers such that $x > y$ . What is the remainder when $2^{x \cdot y + 1}$ is divided by $13$ ?

2

11

3

6

Show answer & explanation

Answer: $2$

Answer

The remainder when

2^{x \cdot y + 1}

is divided by

13

is

2

.

The seven-digit number

75x38y2

is divisible by

72

, which means it must satisfy divisibility by both

8

and

9

. The last three digits

8y2

are divisible by

8

only when

y = 3

or

y = 7

. The sum of digits

25 + x + y

is divisible by

9

when

x + y = 2

or

x + y = 11

. Testing

y = 3

gives

x = 8

, which satisfies

x > y

. Testing

y = 7

gives

x = 4

, which violates

x > y

. Thus,

x = 8

and

y = 3

, giving

x \cdot y + 1 = 25

. By Fermat's Little Theorem,

2^{12} \equiv 1 \pmod{13}

, so

2^{25} = (2^{12})^2 \cdot 2^1 \equiv 2 \pmod{13}

.

Step-by-Step Solution

1

Apply divisibility rule by 8 to find possible values of y

y

can be either

3

or

7

A number is divisible by

72

if it is divisible by both

8

and

9

. For divisibility by

8

, the last three digits

8y2

must be divisible by

8

. Testing single digits gives

832 / 8 = 104

and

872 / 8 = 109

.

2

Apply divisibility rule by 9 to find corresponding x values and enforce x > y

x = 8

and

y = 3

For divisibility by

9

, the sum of digits

(7 + 5 + x + 3 + 8 + y + 2 = 25 + x + y)

must be a multiple of

9

. If

y = 3

,

28 + x = 36 \implies x = 8

, satisfying

x > y

. If

y = 7

,

32 + x = 36 \implies x = 4

, which violates

x > y

.

3

Evaluate the exponent x * y + 1

Exponent = 25

Substituting

x = 8

and

y = 3

gives

8 \cdot 3 + 1 = 25

.

4

Calculate 2^25 mod 13 using Fermat's Little Theorem

Remainder is 2

Since

13

is prime, Fermat's Little Theorem states

2^{12} \equiv 1 \pmod{13}

. Therefore,

2^{25} = (2^{12})^2 \cdot 2^1 \equiv 1^2 \cdot 2 \equiv 2 \pmod{13}

.

Key Concept

Combining composite divisibility rules (8 and 9) with Fermat's Little Theorem for large power remainder evaluation.

Question 230Question →

Calculate the exact numerical value of the following mathematical expression by applying the standard order of operations (BODMAS):

36.4 - \left[ 12.8 + \left\{ 6.5 \times \left( 4.4 - \overline{2.6 + 0.8} \right) \right\} \div 1.3 \right]

Show answer & explanation

Answer: 18.6

Answer

The simplified value of the mathematical expression is 18.6.

Strict application of BODMAS requires simplifying grouping symbols from the innermost vinculum outward, followed by resolving division prior to addition within brackets, yielding an exact answer of 18.6.

Step-by-Step Solution

1

Evaluate the expression underneath the vinculum (bar line)

\overline{2.6 + 0.8} = 3.4

The vinculum functions as an innermost bracket with highest evaluation priority.

2

Evaluate the terms inside the round brackets

4.4 - 3.4 = 1.0

Process operations inside round brackets next.

3

Perform multiplication within the curly braces

6.5 \times 1.0 = 6.5

Complete operations within the curly braces.

4

Execute division inside the square brackets

6.5 \div 1.3 = 5.0

Division takes precedence over addition according to BODMAS rules.

5

Perform addition within the square brackets

12.8 + 5.0 = 17.8

Finish evaluating all operations contained within the square brackets.

6

Perform final subtraction from left to right

36.4 - 17.8 = 18.6

Complete the outermost subtraction to arrive at the final simplified value.

Key Concept

BODMAS Rule with Vinculum and Decimals

Question 231Question →

What is the simplified value of $\sqrt{7 + 4\sqrt{3}} - \sqrt{3}$ ?

Show answer & explanation

Answer: 2

Answer

The simplified value is 2.

Expressing

7 + 4\sqrt{3}

as

(2 + \sqrt{3})^2

allows the square root to simplify directly to

2 + \sqrt{3}

. Subtracting

\sqrt{3}

leaves the exact numerical answer 2.

Step-by-Step Solution

1

Rewrite the expression under the square root as a perfect square of a binomial.

7 + 4\sqrt{3} = 2^2 + (\sqrt{3})^2 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2

Using the identity

(a+b)^2 = a^2 + b^2 + 2ab

, setting

a = 2

and

b = \sqrt{3}

yields

a^2 + b^2 = 4 + 3 = 7

and

2ab = 4\sqrt{3}

.

2

Evaluate the square root of the perfect square.

\sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}

The principal square root of a positive squared expression

\sqrt{x^2}

is

x

.

3

Perform the subtraction indicated in the stem.

(2 + \sqrt{3}) - \sqrt{3} = 2

The radical terms

\sqrt{3}

and

-\sqrt{3}

cancel out, leaving the integer 2.

Key Concept

Simplification of Nested Surds

Question 232Question →

Find the exact numerical value of the following mathematical expression evaluated strictly according to the VBODMAS rule:

75\% \text{ of } 160 - \left[ 3.5 \times 8 + \left\{ 48 \div \left( 14 - \overline{5 \times 2} \right) \right\} \right]

Show answer & explanation

Answer: 80

Answer

The simplified numerical value of the expression is

80

.

Evaluating the expression following strict VBODMAS hierarchy: bar expression

\overline{5 \times 2} = 10

, round brackets

14 - 10 = 4

, curly brackets

48 \div 4 = 12

, square brackets

3.5 \times 8 + 12 = 40

, and percentage 'of' term

75\% \text{ of } 160 = 120

. Finally,

120 - 40 = 80

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{5 \times 2} = 10

According to VBODMAS, operations grouped under a bar take highest priority.

2

Evaluate inside the innermost round brackets

(14 - 10)

14 - 10 = 4

Perform subtraction inside the parentheses.

3

Evaluate inside the curly brackets

\{48 \div 4\}

48 \div 4 = 12

Divide the number outside the round bracket by the result of the round bracket.

4

Evaluate inside the square brackets

[3.5 \times 8 + 12]

28 + 12 = 40

Perform multiplication (

3.5 \times 8 = 28

) before adding

12

.

5

Calculate the percentage term

75\% \text{ of } 160

\frac{75}{100} \times 160 = 120

Evaluate the 'of' operation before final subtraction.

6

Perform final subtraction

120 - 40

80

Subtract the total result of the bracketed expression from the percentage term.

Key Concept

Order of Operations (VBODMAS Rule)

Question 233Question →

Evaluate the following numerical expression strictly according to the VBODMAS rule:

2.5 \times 12 - \left[ 8 + \left\{ 15 \div \left( 4.5 - \overline{2.1 + 0.9} \right) \right\} \right]

What is the final calculated value?

Show answer & explanation

Answer: 12

Answer

12

Evaluating the expression following strict VBODMAS priority yields: Vinculum (2.1 + 0.9 = 3) -> Round brackets (4.5 - 3 = 1.5) -> Curly brackets (15 / 1.5 = 10) -> Square brackets (8 + 10 = 18) -> Multiplication (2.5 * 12 = 30) -> Subtraction (30 - 18 = 12).

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

2.1 + 0.9 = 3

According to the VBODMAS rule, operations under a vinculum take highest priority.

2

Evaluate the innermost round brackets

4.5 - 3 = 1.5

Perform subtraction inside the round brackets after resolving the vinculum.

3

Perform division within the curly brackets

15 / 1.5 = 10

Division inside curly brackets must be calculated before simplifying the outer bracket.

4

Perform addition within the square brackets

8 + 10 = 18

Complete the evaluation of the square bracket section.

5

Perform multiplication outside the brackets

2.5 * 12 = 30

Multiplication takes precedence over final subtraction.

6

Perform final subtraction

30 - 18 = 12

Complete the expression by subtracting the bracket result from the product.

Key Concept

VBODMAS Rule (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Question 234Question →

Find the total number of positive factors of the integer $75$ .

Show answer & explanation

Answer: 6

Answer

The total number of positive factors of 75 is 6.

The prime factorization of 75 is

3^1 \times 5^2

. The number of positive factors is obtained by adding 1 to each exponent in the prime factorization and multiplying the results:

(1 + 1) \times (2 + 1) = 2 \times 3 = 6

. The complete list of positive factors is 1, 3, 5, 15, 25, and 75.

Step-by-Step Solution

1

Express 75 in terms of its prime factors.

75 = 3^1 \times 5^2

Prime factorization breaks down the number into prime components essential for counting factors.

2

Apply the standard formula for total positive factors of a number

N = p^a \times q^b

.

Total factors

= (1 + 1) \times (2 + 1) = 2 \times 3 = 6

Each factor is formed by choosing a power of 3 (2 choices:

3^0, 3^1

) and a power of 5 (3 choices:

5^0, 5^1, 5^2

).

Key Concept

Total Number of Positive Factors from Prime Factorization

Question 235Question →

What is the total number of positive integer factors of $120$ ?

Show answer & explanation

Answer: 16

Answer

The total number of positive integer factors of

120

is

16

.

By prime factorizing

120

, we get

2^3 \times 3^1 \times 5^1

. Applying the factor count formula

(a+1)(b+1)(c+1)

, we obtain

(3+1)(1+1)(1+1) = 4 \times 2 \times 2 = 16

.

Step-by-Step Solution

1

Find the prime factorization of

120

120 = 2^3 \times 3^1 \times 5^1

To calculate the total number of factors systematically, express the number as a product of prime factors.

2

Apply the total factor count formula

Number of factors =

(3 + 1)(1 + 1)(1 + 1)

If a number

N = p^a \times q^b \times r^c

, the total number of positive factors is

(a+1)(b+1)(c+1)

.

3

Compute the product

4 \times 2 \times 2 = 16

Multiplying the incremented prime exponents gives the total count of factors.

Key Concept

Number of Positive Factors from Prime Factorization

Question 236Question →

If $\frac{2^{x+3} \cdot 4^{2x-1}}{8^{x+1}} = 64$ , what is the value of $(x + 2)^2$ ?

Show answer & explanation

Answer: 36

Answer

The value of

(x + 2)^2

is 36.

Converting all terms to base 2 simplifies the equation to

2^{2x-2} = 2^6

. Equating the powers gives

2x - 2 = 6

, so

x = 4

. Substituting

x = 4

into

(x + 2)^2

yields

(4 + 2)^2 = 36

.

Step-by-Step Solution

1

Convert all exponential terms to base 2

Numerator term

4^{2x-1} = 2^{4x-2}

, denominator term

8^{x+1} = 2^{3x+3}

, and right-hand side

64 = 2^6

.

To combine powers using index laws, all expressions must share a common base.

2

Simplify the left-hand side expression using exponent laws

\frac{2^{x+3} \cdot 2^{4x-2}}{2^{3x+3}} = \frac{2^{5x+1}}{2^{3x+3}} = 2^{(5x+1)-(3x+3)} = 2^{2x-2}

.

Apply product law

a^m \cdot a^n = a^{m+n}

and quotient law

\frac{a^m}{a^n} = a^{m-n}

.

3

Solve for the variable x

2^{2x-2} = 2^6 \implies 2x - 2 = 6 \implies x = 4

.

When bases are equal, exponents must be equal.

4

Evaluate the requested target expression

(4 + 2)^2 = 6^2 = 36

.

Substitute the calculated value of

x = 4

into

(x + 2)^2

.

Key Concept

Laws of Indices and Exponential Equations

Estimated Time:1m 30s

Question 237Question →

What is the remainder when $62 \times 53$ is divided by $60$ ?

14

46

55

9

Show answer & explanation

Answer: 46

Answer

46

To find the remainder of a product, we can multiply the remainders of each individual number. When divided by 60, 62 leaves a remainder of 2. So, we multiply 2 by 53 to get 106. Finally, we divide 106 by 60, which leaves a remainder of 46.

Step-by-Step Solution

1

Find the remainder of each factor when divided by 60.

62 \equiv 2 \pmod{60}

and

53 \equiv 53 \pmod{60}

(or

-7 \pmod{60}

).

By the properties of modular arithmetic, we can simplify large multiplications by replacing numbers with their remainders.

2

Multiply the remainders together.

2 \times 53 = 106

. Alternatively, using negative remainders:

2 \times (-7) = -14

.

The remainder of a product is equal to the product of the individual remainders.

3

Find the final remainder by dividing the product from Step 2 by 60.

106 = 60 \times 1 + 46

, so the remainder is

46

. (If using

-14

, add

60

to get

46

).

The intermediate product must be reduced modulo 60 to find the final positive remainder.

Key Concept

Properties of Remainders in Multiplication

Question 238Question →

Consider the number obtained by multiplying $48$ and $53$ . If this product is divided by $50$ , what is the resulting positive remainder?

Show answer & explanation

Answer: 44

Answer

44

Applying the properties of modular arithmetic,

48

is congruent to

-2 \pmod{50}

and

53

is congruent to

3 \pmod{50}

. Multiplying these gives a remainder of

-6 \pmod{50}

. To find the standard positive remainder, we add the divisor

50

to

-6

, resulting in

44

. Alternatively, direct calculation gives

48 \times 53 = 2544

, and

2544

divided by

50

yields a quotient of

50

with a remainder of

44

.

Step-by-Step Solution

1

Find the remainder of 48 divided by 50.

-2

Using a negative remainder simplifies the multiplication step.

2

Find the remainder of 53 divided by 50.

3

Individual remainders are needed to apply the product rule of modular arithmetic.

3

Multiply the individual remainders.

-2 \times 3 = -6

The remainder of a product is congruent to the product of the individual remainders.

4

Convert the negative remainder to a positive remainder.

50 - 6 = 44

A standard remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Remainder Theorem and Negative Modulus

Question 239Question →

If the equation $\sqrt{x} + \sqrt{x - 9} = 9$ holds true, what is the exact value of $x$ ?

Show answer & explanation

Answer: 25

Answer

25

The exact value of

x

is 25. By moving one radical to the right side and squaring both sides, we eliminate one square root. Simplifying and isolating the remaining square root allows us to square both sides a second time, revealing the final value. Alternatively, using the conjugate property of surds: multiplying both sides of the identity

(\sqrt{x})^2 - (\sqrt{x-9})^2 = 9

by their difference gives

(\sqrt{x} - \sqrt{x-9})(\sqrt{x} + \sqrt{x-9}) = 9

. Since the sum is 9, the difference must be 1 (i.e.,

\sqrt{x} - \sqrt{x-9} = 1

). Adding this back to the original equation yields

2\sqrt{x} = 10

, so

\sqrt{x} = 5

and

x = 25

.

Step-by-Step Solution

1

Isolate one of the square root terms on one side of the equation.

\sqrt{x} = 9 - \sqrt{x - 9}

Isolating a radical makes it easier to eliminate it by squaring both sides.

2

Square both sides of the equation and expand the right side.

x = 81 - 18\sqrt{x - 9} + (x - 9)

Squaring eliminates the isolated radical. The right side is expanded using the algebraic identity

(a - b)^2 = a^2 - 2ab + b^2

.

3

Simplify the equation by canceling

x

from both sides and combining constant terms.

x = x + 72 - 18\sqrt{x - 9} \Rightarrow 18\sqrt{x - 9} = 72

Combining like terms simplifies the equation, leaving only a single radical expression.

4

Divide by 18 and square both sides one final time to solve for

x

.

\sqrt{x - 9} = 4 \Rightarrow x - 9 = 16 \Rightarrow x = 25

Isolating the final radical and squaring removes the remaining root, yielding a simple linear equation for

x

.

Key Concept

Solving radical equations and applying algebraic identities with surds.

Question 240Question →

Suppose a certain positive integer $x$ yields a remainder of $4$ upon division by $17$ . What is the remainder obtained when the quantity $5x - 38$ is divided by $17$ ?

-1

0

1

16

Show answer & explanation

Answer: $16$

Answer

The correct answer is

16

.

By substituting the initial remainder

4

into the expression, we evaluate

5(4) - 38 = -18

. To find the valid positive remainder modulo

17

, we add multiples of

17

until the number becomes non-negative:

-18 + 34 = 16

.

Step-by-Step Solution

1

Identify the modular relationship for

x

x \equiv 4 \pmod{17}

The problem states that

x

leaves a remainder of

4

when divided by

17

.

2

Substitute this remainder into the given expression

5x - 38

5(4) - 38 = 20 - 38 = -18

Using the properties of modular arithmetic, we can substitute the remainder directly into polynomial expressions.

3

Find the equivalent positive remainder for

-18

modulo

17

-18 + 17 + 17 = 16

Remainders must be non-negative integers strictly less than the divisor. Adding multiples of

17

to

-18

gives the valid positive equivalent.

Key Concept

Modular Arithmetic and Negative Remainders

Basic Numeracy

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