Basic Numeracy Practice Questions — State PSC Exam — Page 11

Calculate the exact numerical value of the given mathematical expression using the VBODMAS rule:

\frac{3}{5} \text{ of } 250 - \left[ 42 + \left\{ 24 \div \left( 7.5 - \overline{4.2 + 0.3} \right) \right\} \right]

Show answer & explanation

Answer: 100

Answer

100

Evaluating strictly according to VBODMAS order of operations yields 100: Vinculum gives 4.5, round brackets give 3, division in curly brackets gives 8, addition in square brackets gives 50, 'of' operation gives 150, and 150 - 50 = 100.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar line)

4.2 + 0.3 = 4.5

According to the VBODMAS rule, the vinculum takes precedence over standard brackets and operations.

2

Simplify the innermost round brackets

7.5 - 4.5 = 3

Operations inside round parentheses ( ) are resolved next.

3

Evaluate the expression inside the curly brackets

24 / 3 = 8

Perform the division operation inside curly braces { }.

4

Evaluate the expression inside the square brackets

42 + 8 = 50

Perform addition within the outer square brackets [ ].

5

Calculate the 'of' operation

(3/5) * 250 = 150

The 'of' operation takes priority over standard addition and subtraction outside brackets.

6

Perform final subtraction

150 - 50 = 100

Subtract the total evaluated bracketed quantity from the 'of' calculation result.

Key Concept

Hierarchical priority of operations in VBODMAS involving vinculum, nested brackets, and fractional 'of' operations

Question 202Question →

A positive integer $N$ has exactly $15$ positive factors and is divisible by $6$ . If $N$ has exactly two distinct prime factors, what is the sum of all possible values of $N$ that are less than $500$ ?

144

324

468

540

Show answer & explanation

Answer: 468

Answer

The sum of all possible values of N less than 500 is 468.

For a number

N = p^a \times q^b

to have 15 factors,

(a+1)(b+1) = 15

. The factor pairs of 15 for two prime factors are

(5, 3)

and

(3, 5)

, which correspond to exponent pairs

(4, 2)

and

(2, 4)

. Since

N

is divisible by 6, its prime factors are 2 and 3. Computing both cases gives

2^4 \times 3^2 = 144

and

2^2 \times 3^4 = 324

. Both are less than 500, making their sum

144 + 324 = 468

.

Step-by-Step Solution

1

Determine the prime factors of N

N has prime factors 2 and 3

Since N is divisible by 6, it must have at least 2 and 3 as prime factors. The problem states N has exactly two distinct prime factors, so its prime factorization is of the form

N = 2^a \times 3^b

.

2

Apply the total factors formula

(a + 1)(b + 1) = 15

The total number of positive factors of

N = p^a \times q^b

is given by

(a + 1)(b + 1) = 15

.

3

Find valid non-negative integer pairs (a, b)

Either

(a=4, b=2)

or

(a=2, b=4)

Since 15 factors into

5 \times 3

or

3 \times 5

(note that

15 \times 1

would imply only one prime factor, which contradicts having two distinct prime factors), the possible exponent pairs are

(4, 2)

and

(2, 4)

.

4

Calculate the values of N and check bounds

N_1 = 2^4 \times 3^2 = 144

and

N_2 = 2^2 \times 3^4 = 324

Both

144 < 500

and

324 < 500

meet all conditions.

5

Sum the valid values of N

144 + 324 = 468

Adding the two valid integers yields the total sum.

Key Concept

Factors and Prime Factorization Exponent Rule

Question 203Question →

Three church bells toll together at intervals of $9$ minutes, $12$ minutes, and $15$ minutes respectively. If they all toll together at $8:00\text{ AM}$ , at what time will they next toll together?

8:03 AM

8:36 AM

10:00 AM

11:00 AM

Show answer & explanation

Answer: 11:00 AM

Answer

11:00 AM

The bells will toll together after a duration equal to the Least Common Multiple (LCM) of their individual tolling intervals. The prime factorizations are

9 = 3^2

,

12 = 2^2 \times 3

, and

15 = 3 \times 5

. Taking the highest power of each prime factor yields

\text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 180

minutes, which is equal to

3

hours. Adding

3

hours to

8:00\text{ AM}

gives

11:00\text{ AM}

.

Step-by-Step Solution

1

Find prime factorizations of each interval

9 = 3^2

,

12 = 2^2 \times 3

, and

15 = 3 \times 5

To compute the Least Common Multiple (LCM), express each number in prime factor form.

2

Calculate the LCM

\text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\text{ minutes}

Take the highest power of each prime factor involved.

3

Convert minutes to hours and add to the initial time

180\text{ minutes} = 3\text{ hours}

;

8:00\text{ AM} + 3\text{ hours} = 11:00\text{ AM}

Determine the exact time the bells will next chime together.

Key Concept

Application of Least Common Multiple (LCM) in Simultaneous Periodic Events

Question 204Question →

What is the simplified numerical value of the following mathematical expression when evaluated strictly according to the VBODMAS rule?

40\% \text{ of } 150 - \left[ 16 + \left\{ 20 \div \left( 8.5 - \overline{3.1 + 1.4} \right) \right\} \right]

39

51

49

40

Show answer & explanation

Answer: 39

Answer

39

Evaluating strictly by VBODMAS priority: first the vinculum yields 4.5, then the round bracket gives 8.5 - 4.5 = 4, then division yields 20 ÷ 4 = 5, followed by square bracket addition 16 + 5 = 21. Finally, subtracting 21 from 40% of 150 (which is 60) gives 60 - 21 = 39.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{3.1 + 1.4} = 4.5

The vinculum has the highest priority and must be evaluated first.

2

Evaluate the terms inside the round brackets

8.5 - 4.5 = 4

The next priority is resolving the innermost parentheses.

3

Perform division inside the curly brackets

20 \div 4 = 5

Inside curly brackets, division takes precedence over addition.

4

Evaluate the terms inside the square brackets

16 + 5 = 21

Resolve the outer square bracket by adding the terms.

5

Calculate the percentage term

40\% \text{ of } 150 = \frac{40}{100} \times 150 = 60

'Of' represents multiplication in percentage operations.

6

Perform final subtraction

60 - 21 = 39

Subtract the total bracket value from the percentage calculation.

Key Concept

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

Question 205Question →

Consider the positive integer $N = 720$ , which has a prime factorization of $2^4 \times 3^2 \times 5^1$ . Which of the following statements regarding the positive factors of $N$ are correct?

Select all that apply

The number of positive factors of

N

that are perfect squares is

6

.

The sum of all positive even factors of

N

is

2340

.

The number of positive factors of

N

that are multiples of

15

is

6

.

The product of all positive factors of

N

is equal to

720^{15}

.

Show answer & explanation

Answer: The number of positive factors of $N$ that are perfect squares is $6$ .; The sum of all positive even factors of $N$ is $2340$ .; The product of all positive factors of $N$ is equal to $720^{15}$ .

Answer

The correct statements are those asserting that

N

has 6 perfect square factors, that the sum of all positive even factors of

N

is 2340, and that the product of all positive factors of

N

is

720^{15}

.

The statements confirming 6 perfect square factors, a sum of 2340 for even factors, and a factor product of

720^{15}

are all mathematically correct applications of prime factorization principles.

Step-by-Step Solution

1

Determine perfect square factors of

N = 2^4 \times 3^2 \times 5^1

For a factor

2^a \times 3^b \times 5^c

to be a square, exponents

a, b, c

must be even. Possible values:

a \in \{0, 2, 4\}

(3 options),

b \in \{0, 2\}

(2 options),

c \in \{0\}

(1 option). Number of square factors

= 3 \times 2 \times 1 = 6

.

Perfect squares require all prime factors to have even exponents.

2

Calculate the sum of all positive even factors

Sum of all factors

= (2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2)(5^0+5^1) = 31 \times 13 \times 6 = 2418

. Sum of odd factors (only

2^0

)

= 1 \times 13 \times 6 = 78

. Sum of even factors

= 2418 - 78 = 2340

.

Even factors are obtained by subtracting odd factor sum from total factor sum.

3

Count factors divisible by 15

Since

15 = 3^1 \times 5^1

, factors of 720 divisible by 15 require

a \in \{0, 1, 2, 3, 4\}

(5 choices),

b \in \{1, 2\}

(2 choices), and

c \in \{1\}

(1 choice). Total factors

= 5 \times 2 \times 1 = 10

.

Divisibility by 15 requires at least power 1 for both prime factors 3 and 5.

4

Compute the product of all positive factors

Total factors

T = (4+1)(2+1)(1+1) = 30

. Product of factors

= N^{T/2} = 720^{30/2} = 720^{15}

.

Factors pair up such that

f_i \times f_{T+1-i} = N

, giving

T/2

pairs.

Key Concept

Properties of positive factors derived from prime factorization: square factor counts, even factor sums, constrained divisibility factor counts, and total factor product formula.

Question 206Question →

What is the simplified numerical value of the following mathematical expression when evaluated strictly using the BODMAS rule?

50 - \left[ 8 + \left\{ 16 \div \left( 8 - \overline{7 - 3} \right) \right\} \right]

38

44

50

46

Show answer & explanation

Answer: 38

Answer

The simplified numerical value of the expression is 38.

Evaluating step-by-step according to VBODMAS order gives: first vinculum 7 - 3 = 4, then round brackets 8 - 4 = 4, followed by curly bracket division 16 ÷ 4 = 4, then square bracket addition 8 + 4 = 12, and finally 50 - 12 = 38. The value 38 correctly demonstrates strict adherence to order of operations.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\overline{7 - 3} = 4

According to VBODMAS rules, operations under a vinculum bar have the highest priority.

2

Evaluate the expression inside the round brackets

8 - 4 = 4

Substitute the vinculum result into the round bracket.

3

Perform division inside the curly brackets

16 \div 4 = 4

Curly brackets are evaluated next after simplifying the inner round brackets.

4

Perform addition inside the square brackets

8 + 4 = 12

Resolve the remaining operations inside the outermost square brackets.

5

Perform final subtraction outside the brackets

50 - 12 = 38

Complete the evaluation by subtracting the bracket value from 50.

Key Concept

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

Question 207Question →

A positive integer $N$ has a prime factorization of the form $p^a \cdot q^b$ , where $p$ and $q$ are distinct prime numbers, and $a, b \ge 1$ . If the total number of positive factors of $N^2$ is $45$ and the total number of positive factors of $N^3$ is $91$ , what is the total number of positive factors of $N$ ?

12

15

18

20

Show answer & explanation

Answer: $15$

Answer

The total number of positive factors of

N

is

15

.

For a composite integer

N = p^a q^b

, the number of positive factors is

(a+1)(b+1)

. Given that

N^2 = p^{2a} q^{2b}

has

45

positive factors, we have

(2a+1)(2b+1) = 45

. Factoring

45

into odd pairs yields two possibilities for

(a, b)

:

(1, 7)

or

(2, 4)

. Using the second condition that

N^3 = p^{3a} q^{3b}

has

91

factors, we check

(3a+1)(3b+1) = 91

. For

(1, 7)

,

(4)(22) = 88 \neq 91

. For

(2, 4)

,

(7)(13) = 91

, which is correct. Therefore,

a=2

and

b=4

, so the number of positive factors of

N

is

(2+1)(4+1) = 15

.

Step-by-Step Solution

1

Set up the factor count equation for

N^2

Since

N = p^a q^b

,

N^2 = p^{2a} q^{2b}

. The number of positive factors of

N^2

is

(2a+1)(2b+1) = 45

.

For a prime factorization

p^x q^y

, the total number of positive factors is

(x+1)(y+1)

.

2

Find possible integer solutions for

(a, b)

The odd factor pairs of

45

are

(3, 15)

and

(5, 9)

. This gives two possible sets of exponents: Case 1:

2a+1=3 \implies a=1

and

2b+1=15 \implies b=7

. Case 2:

2a+1=5 \implies a=2

and

2b+1=9 \implies b=4

.

Since

a, b \ge 1

, both factors

2a+1

and

2b+1

must be odd integers greater than

1

.

3

Evaluate candidate solutions using the second condition for

N^3

For

N^3 = p^{3a} q^{3b}

, the number of factors is

(3a+1)(3b+1) = 91

. Testing Case 1 (

a=1, b=7

):

(3(1)+1)(3(7)+1) = 4 \times 22 = 88 \neq 91

. Testing Case 2 (

a=2, b=4

):

(3(2)+1)(3(4)+1) = 7 \times 13 = 91

, which matches.

The true exponent values must simultaneously satisfy both given conditions.

4

Calculate the total number of positive factors of

N

With

a=2

and

b=4

,

N = p^2 q^4

. The total number of positive factors of

N

is

(2+1)(4+1) = 3 \times 5 = 15

.

Apply the total factor formula

(a+1)(b+1)

to

N

.

Key Concept

Calculating factor counts of a prime-factored number from equations involving factor counts of its powers

Estimated Time:2m 0s

Question 208Question →

A positive integer $N$ is divisible by $10$ . The sum of all distinct prime factors of $N$ is $10$ , and $N$ has exactly $12$ positive factors. What is the smallest possible value of $N$ ?

Show answer & explanation

Answer: 60

Answer

The smallest possible value of

N

is

60

.

Because

N

is divisible by

10

, its prime factorization must contain

2

and

5

. The sum of all distinct prime factors is

10

, which requires

2 + 5 + p = 10

, giving

p = 3

. Hence,

N = 2^a \times 3^b \times 5^c

with

a, b, c \ge 1

. The number of factors is

(a+1)(b+1)(c+1) = 12

. The unique product representation of

12

using three integers

\ge 2

is

3 \times 2 \times 2

, meaning the exponents are one

2

and two

1

s. Minimizing

N

requires assigning the highest exponent

2

to the smallest prime

2

, giving

N = 2^2 \times 3^1 \times 5^1 = 60

.

Step-by-Step Solution

1

Identify the distinct prime factors of

N

The distinct prime factors are

2, 3,

and

5

.

Divisibility by

10

implies

2

and

5

are prime factors of

N

. The sum of distinct prime factors is

10

, so the remaining prime factor is

10 - (2 + 5) = 3

.

2

Determine the exponents in the prime factorization of

N

The set of exponents

\{a, b, c\}

must be

\{2, 1, 1\}

.

The total number of factors is

(a+1)(b+1)(c+1) = 12

. Since

a, b, c \ge 1

, the only integer partition of

12

into three factors

\ge 2

is

3 \times 2 \times 2

.

3

Assign exponents to minimize the value of

N

N = 2^2 \times 3^1 \times 5^1 = 60

.

To minimize

N

, the largest exponent

2

must be assigned to the smallest prime base

2

.

Key Concept

Prime factorization, distinct prime factor properties, and total factor count formula

Question 209Question →

What is the simplified value of the following mathematical expression when evaluated strictly using the VBODMAS rule?

15 + 2.5 \times \left[ 16 - \left\{ 10 \div \left( 3.5 - \overline{2.1 - 0.6} \right) \right\} \right]

42.5

192.5

23.75

50

Show answer & explanation

Answer: 42.5

Answer

The simplified value of the expression is 42.5.

Evaluating strictly according to VBODMAS rules, we first simplify the expression under the vinculum bar (2.1 - 0.6 = 1.5), then the round brackets (3.5 - 1.5 = 2), followed by the curly brackets (10 ÷ 2 = 5), and then the square brackets (16 - 5 = 11). Finally, executing multiplication before addition yields 15 + (2.5 × 11) = 15 + 27.5 = 42.5.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{2.1 - 0.6} = 1.5

According to VBODMAS, the vinculum takes highest priority over standard brackets.

2

Evaluate the terms inside the innermost parentheses (round brackets)

3.5 - 1.5 = 2.0

Parentheses are evaluated next after resolving the vinculum.

3

Evaluate the expression inside the curly brackets

10 \div 2.0 = 5

Division within the curly brackets takes priority before moving outward.

4

Evaluate the expression inside the square brackets

16 - 5 = 11

Resolve the remaining operations within the square brackets.

5

Perform multiplication before addition

15 + (2.5 \times 11) = 15 + 27.5 = 42.5

Multiplication takes precedence over addition according to BODMAS rules.

Key Concept

Order of Operations (VBODMAS Rule)

Question 210Question →

If $5^{x+1} + 5^{x-1} = 650$ , what is the value of $2^{2x - 1}$ ?

32

64

16

8

Show answer & explanation

Answer: $32$

Answer

32

Factoring out

5^{x-1}

yields

26 \cdot 5^{x-1} = 650

, which simplifies to

5^{x-1} = 25

, giving

x = 3

. Substituting

x = 3

into

2^{2x - 1}

gives

2^5 = 32

.

Step-by-Step Solution

1

Factor out common exponential terms from the given equation.

5^{x-1}(5^2 + 1) = 650 \implies 5^{x-1}(25 + 1) = 650 \implies 26 \cdot 5^{x-1} = 650

Use the product law of indices

5^{x+1} = 5^{x-1} \cdot 5^2

to rewrite the expression.

2

Solve for the unknown variable

x

.

5^{x-1} = \frac{650}{26} = 25 = 5^2 \implies x - 1 = 2 \implies x = 3

Since the bases are identical, equate the exponents.

3

Substitute

x = 3

into the expression

2^{2x - 1}

.

2^{2(3) - 1} = 2^{6 - 1} = 2^5 = 32

Evaluate the power after computing the exponent value.

Key Concept

Solving Exponential Equations using Laws of Indices

Question 211Question →

If $x = \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} - \sqrt{5}}$ and $y = \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} + \sqrt{5}}$ , what is the value of $x^2 + y^2 + xy$ ?

142

143

144

145

Show answer & explanation

Answer: $143$

Answer

The value of

x^2 + y^2 + xy

is

143

.

Rationalizing both surds gives

x = 6 + \sqrt{35}

and

y = 6 - \sqrt{35}

. The sum

x + y = 12

and product

xy = 1

. Substituting into the algebraic identity

x^2 + y^2 + xy = (x + y)^2 - xy

gives

12^2 - 1 = 143

.

Step-by-Step Solution

1

Rationalize the denominator for

x

and

y

x = \frac{(\sqrt{7} + \sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} = \frac{7 + 5 + 2\sqrt{35}}{2} = 6 + \sqrt{35}

, and similarly

y = 6 - \sqrt{35}

.

Eliminating radicals from denominators simplifies calculations.

2

Compute the sum

(x + y)

and product

xy

x + y = (6 + \sqrt{35}) + (6 - \sqrt{35}) = 12

, and

xy = (6 + \sqrt{35})(6 - \sqrt{35}) = 36 - 35 = 1

.

Using symmetric expressions simplifies evaluating degree 2 polynomials.

3

Express

x^2 + y^2 + xy

in terms of

(x + y)

and

xy

x^2 + y^2 + xy = (x + y)^2 - xy = 12^2 - 1 = 144 - 1 = 143

.

Applying the identity

x^2 + y^2 = (x + y)^2 - 2xy

gives

x^2 + y^2 + xy = (x + y)^2 - xy

.

Key Concept

Rationalization of surds and application of algebraic identities

Question 212Question →

If $\frac{3^{2m + 1} \cdot 9^{m - 2}}{27^{m - 1}} = 81$ , what is the value of $m$ ?

Show answer & explanation

Answer: 4

Answer

The value of

m

is 4.

By converting all terms in the equation to base 3 and applying index rules (

a^x \cdot a^y = a^{x+y}

and

\frac{a^x}{a^y} = a^{x-y}

), the expression simplifies directly to

3^m = 3^4

, leading to

m = 4

.

Step-by-Step Solution

1

Express all terms with a common base of 3

9^{m-2} = (3^2)^{m-2} = 3^{2m-4}

and

27^{m-1} = (3^3)^{m-1} = 3^{3m-3}

Rewriting powers with a single prime base allows for simplification using the laws of indices.

2

Simplify the numerator using the product rule of exponents

3^{2m+1} \cdot 3^{2m-4} = 3^{(2m+1) + (2m-4)} = 3^{4m-3}

According to the product rule

a^x \cdot a^y = a^{x+y}

, exponents with the same base are added during multiplication.

3

Apply the quotient rule of exponents to simplify the left-hand side

\frac{3^{4m-3}}{3^{3m-3}} = 3^{(4m-3) - (3m-3)} = 3^m

According to the quotient rule

\frac{a^x}{a^y} = a^{x-y}

, exponents are subtracted during division.

4

Express the right-hand side in base 3 and solve for

m

3^m = 81 = 3^4 \implies m = 4

Since the bases on both sides of the equation are equal, the exponents must also be equal.

Key Concept

Exponential Equations and Laws of Indices

Question 213Question →

If $\left(\sqrt{\frac{3}{5}}\right)^{x + 1} = \left(\frac{125}{27}\right)^{x - 1}$ , what is the value of $x$ ?

\frac{5}{7}

\frac{7}{5}

\frac{1}{2}

\frac{1}{3}

Show answer & explanation

Answer: $\frac{5}{7}$

Answer

\frac{5}{7}

By converting both sides of the equation to the common base

\frac{3}{5}

, the left side becomes

\left(\frac{3}{5}\right)^{\frac{x+1}{2}}

and the right side becomes

\left(\frac{3}{5}\right)^{-3(x-1)}

. Equating exponents gives

\frac{x+1}{2} = 3 - 3x

, which simplifies directly to

x = \frac{5}{7}

.

Step-by-Step Solution

1

Express the left side of the equation using fractional exponents

\left(\sqrt{\frac{3}{5}}\right)^{x+1} = \left(\left(\frac{3}{5}\right)^{\frac{1}{2}}\right)^{x+1} = \left(\frac{3}{5}\right)^{\frac{x+1}{2}}

The square root of a quantity corresponds to an exponent of

\frac{1}{2}

.

2

Express the right side with base

\frac{3}{5}

\left(\frac{125}{27}\right)^{x-1} = \left(\left(\frac{5}{3}\right)^3\right)^{x-1} = \left(\left(\frac{3}{5}\right)^{-3}\right)^{x-1} = \left(\frac{3}{5}\right)^{-3(x-1)}

Since

125 = 5^3

and

27 = 3^3

,

\frac{125}{27} = \left(\frac{5}{3}\right)^3 = \left(\frac{3}{5}\right)^{-3}

.

3

Equate the exponents since the bases are identical

\frac{x+1}{2} = -3(x-1) \implies \frac{x+1}{2} = 3 - 3x

If

a^m = a^n

for

a > 0

and

a \neq 1

, then

m = n

.

4

Solve the linear equation for

x

x + 1 = 2(3 - 3x) \implies x + 1 = 6 - 6x \implies 7x = 5 \implies x = \frac{5}{7}

Standard algebraic simplification to isolate

x

.

Key Concept

Laws of Indices and Rational Base Equivalence

Question 214Question →

If $x = 7 - 4\sqrt{3}$ , what is the value of $\sqrt{x} + \frac{1}{\sqrt{x}}$ ?

Show answer & explanation

Answer: 4

Answer

The value of

\sqrt{x} + \frac{1}{\sqrt{x}}

is 4.

Expressing

7 - 4\sqrt{3}

as

(2 - \sqrt{3})^2

allows taking the square root to get

\sqrt{x} = 2 - \sqrt{3}

. Rationalizing its reciprocal gives

\frac{1}{\sqrt{x}} = 2 + \sqrt{3}

. Adding these two values cancels the irrational component

\sqrt{3}

, leaving

2 + 2 = 4

.

Step-by-Step Solution

1

Simplify the nested surd

\sqrt{x} = \sqrt{7 - 4\sqrt{3}}

\sqrt{x} = 2 - \sqrt{3}

Rewrite

7 - 4\sqrt{3}

as

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

and take the principal square root.

2

Calculate the reciprocal

\frac{1}{\sqrt{x}}

by rationalizing the denominator

\frac{1}{\sqrt{x}} = 2 + \sqrt{3}

Multiply numerator and denominator of

\frac{1}{2 - \sqrt{3}}

by its conjugate

(2 + \sqrt{3})

.

3

Add

\sqrt{x}

and

\frac{1}{\sqrt{x}}

4

Sum

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

so that the radical terms cancel out.

Key Concept

Simplification of surds of the form

\sqrt{a \pm \sqrt{b}}

and rationalization using conjugates

Question 215Question →

What is the remainder when the value of the expression $2^{40} - 5 \times 3$ is divided by $10$ ?

9

1

3

6

Show answer & explanation

Answer: 1

Answer

1

The remainder when dividing by 10 is determined by the unit digit of the expression. First, evaluate the multiplication:

5 \times 3 = 15

. Next, find the unit digit of

2^{40}

. The powers of 2 follow a cyclicity of 4 (2, 4, 8, 6). Since 40 is perfectly divisible by 4, the unit digit corresponds to the 4th value in the cycle, which is 6. Substituting these into the expression gives

6 - 15 = -9

. To find the valid positive remainder modulo 10, the divisor 10 must be added to the negative result:

-9 + 10 = 1

.

Step-by-Step Solution

1

Evaluate the multiplication part of the expression.

The term

5 \times 3

equals

15

. The expression simplifies to

2^{40} - 15

.

The order of operations (BODMAS) requires multiplication to be performed before subtraction.

2

Determine the unit digit (remainder modulo 10) of

2^{40}

by identifying the cyclicity of base 2.

The unit digits of powers of 2 repeat in a cycle of 4: 2, 4, 8, 6.

Since dividing by 10 is equivalent to finding the unit digit, we use power cyclicity to simplify large exponents.

3

Divide the exponent 40 by the cyclicity length 4.

40 divided by 4 leaves a remainder of 0. A remainder of 0 corresponds to the 4th position in the cycle.

The remainder determines the position in the cyclicity sequence. A remainder of 0 means the sequence has completed a full cycle.

4

Substitute the determined unit digit back into the simplified expression.

The 4th power in the cycle ends in 6. Substituting this into the expression yields

6 - 15 = -9

.

Replacing the large power with its unit digit equivalent allows us to calculate the raw remainder.

5

Convert the negative remainder into a valid positive remainder modulo 10.

-9 + 10 = 1

. The final remainder is 1.

Remainders must be non-negative integers smaller than the divisor. Adding the divisor to a negative remainder provides the correct positive equivalent.

Key Concept

Applying cyclicity rules, the remainder theorem, and the order of operations to evaluate complex numerical expressions.

Question 216Question →

If $2^{x+1} + 2^{x-1} = 320$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x

is 7.

Factoring out

2^{x-1}

from the expression gives

2^{x-1}(2^2 + 1) = 320

, which simplifies to

5 \cdot 2^{x-1} = 320

. Dividing by 5 yields

2^{x-1} = 64 = 2^6

. Equating exponents gives

x - 1 = 6

, so

x = 7

.

Step-by-Step Solution

1

Rewrite terms using exponent rules to factor out the common power

2^{x-1}

2^{x-1}(2^2 + 1) = 320

Using the product rule

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

, we have

2^{x+1} = 2^{x-1} \cdot 2^2

.

2

Simplify the numerical factor inside the parentheses

5 \cdot 2^{x-1} = 320

2^2 + 1 = 4 + 1 = 5

.

3

Isolate the exponential term by dividing by 5

2^{x-1} = 64

Dividing

320

by

5

gives

64

.

4

Write 64 as a power of 2 and solve for

x

2^{x-1} = 2^6 \implies x - 1 = 6 \implies x = 7

Equating exponents when bases are equal.

Key Concept

Solving Exponential Equations by Factoring Common Powers

Question 217Question →

What is the remainder when the sum

22^{2026} + 39^{2026} + 56^{2026} + 73^{2026} + 90^{2026}

is divided by

17

?

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Answer: 11

Answer

11

By reducing all bases modulo 17, the sum becomes five identical terms of

5^{2026}

, which simplifies to

5^{2027}

. Using Fermat's Little Theorem (

5^{16} \equiv 1 \pmod{17}

), the exponent 2027 is reduced modulo 16 to 11. Finally, calculating

5^{11} \pmod{17}

yields

-40

, which corresponds to a positive remainder of 11.

Step-by-Step Solution

1

Reduce each base in the expression modulo 17.

The expression simplifies to

5^{2026} + 5^{2026} + 5^{2026} + 5^{2026} + 5^{2026} \pmod{17}

.

Using the property

(A \pmod M)^k \pmod M = A^k \pmod M

, we find that 22, 39, 56, 73, and 90 all leave a remainder of 5 when divided by 17.

2

Combine the identical terms into a single expression.

The sum becomes

5 \times 5^{2026} = 5^{2027} \pmod{17}

.

Since there are exactly 5 identical terms, adding them together is equivalent to multiplying the term by 5, which conveniently increases the exponent by 1.

3

Apply Fermat's Little Theorem to reduce the large exponent.

Fermat's Little Theorem states

5^{16} \equiv 1 \pmod{17}

. Dividing the exponent 2027 by 16 yields

2027 = 16 \times 126 + 11

.

Because 17 is a prime number and does not divide 5, we can use

a^{p-1} \equiv 1 \pmod p

to eliminate full cycles of 16 in the exponent.

4

Simplify the remaining expression

5^{11} \pmod{17}

.

The expression reduces to

-6 \pmod{17}

.

We break down the power:

5^2 = 25 \equiv 8

,

5^4 \equiv 8^2 = 64 \equiv -4

, and

5^8 \equiv (-4)^2 = 16 \equiv -1

. Thus,

5^{11} = 5^8 \times 5^2 \times 5^1 \equiv (-1) \times 8 \times 5 = -40

. Since

-40 = -3 \times 17 + 11

, the remainder is 11 (or

-6

).

5

Convert any negative remainder to a valid positive remainder.

-6 + 17 = 11

. The final remainder is 11.

A standard remainder must be a positive integer strictly less than the divisor.

Key Concept

Fermat's Little Theorem and Modulo Arithmetic Reductions

Question 218Question →

What is the smallest positive integer that has exactly $21$ positive divisors?

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Answer: 576

Answer

The smallest positive integer with exactly 21 positive divisors is 576.

The number 576 factors completely into

2^6 \times 3^2

. According to the divisor formula, we take the exponents, add 1 to each, and multiply them:

(6+1) \times (2+1) = 7 \times 3 = 21

. Because 2 is the smallest prime and is paired with the largest exponent, 576 is the absolute minimum positive integer that can produce exactly 21 divisors.

Step-by-Step Solution

1

Use the divisor counting formula.

The number of divisors of

N = p^a \times q^b

is

(a+1)(b+1)

. We set this equal to

21

.

This establishes the relationship between prime exponents and the given divisor count.

2

Find the factor pairs of 21.

The ways to factor 21 are

21

(as

21 \times 1

) or

7 \times 3

.

This determines the possible combinations of

(a+1)

and

(b+1)

.

3

Test the first case (single prime factor).

a+1 = 21 \implies a = 20

. The smallest number is

2^{20} = 1,048,576

.

We must evaluate all cases to ensure we find the absolute minimum.

4

Test the second case (two prime factors).

(a+1)(b+1) = 7 \times 3 \implies a = 6, b = 2

. The number form is

p^6 \times q^2

.

This evaluates the only other valid prime exponent combination.

5

Minimize the two-prime case.

Assign

p=2

and

q=3

to get

2^6 \times 3^2 = 64 \times 9 = 576

.

Assigning the smallest available prime to the largest exponent minimizes the product.

6

Compare the results of both cases.

576 < 1,048,576

.

To conclusively identify the smallest positive integer.

Key Concept

Prime Factorization and the Divisor Counting Formula

Question 219Question →

What is the positive remainder when the expression $12^{77} - 26^{77}$ is divided by $19$ ?

9

16

5

3

Show answer & explanation

Answer: 16

Answer

The correct positive remainder is 16.

By reducing the bases modulo 19, we get

12 \equiv -7

and

26 \equiv 7

. The expression transforms to

(-7)^{77} - 7^{77}

, which simplifies to

-2 \times 7^{77}

since the exponent is odd. Finding the cycle of

7 \pmod{19}

reveals that

7^3 \equiv 1 \pmod{19}

. Because

77 = 3 \times 25 + 2

, we have

7^{77} \equiv 7^2 \equiv 11 \pmod{19}

. Finally,

-2 \times 11 = -22

, and adjusting this to a positive remainder by adding 38 (a multiple of 19) gives 16.

Step-by-Step Solution

1

Express the bases of the exponents in terms of their remainders when divided by 19.

12 \equiv -7 \pmod{19}

and

26 \equiv 7 \pmod{19}

.

Converting large bases to smaller equivalent modular values (especially symmetric ones) simplifies the exponentiation.

2

Substitute the simplified bases back into the original expression.

12^{77} - 26^{77} \equiv (-7)^{77} - 7^{77} \pmod{19}

.

This reduces the problem to calculating powers of 7.

3

Simplify the expression by handling the negative sign.

Since 77 is an odd number,

(-7)^{77} = -(7^{77})

. The expression becomes

-(7^{77}) - 7^{77} = -2 \times 7^{77} \pmod{19}

.

Factoring out

7^{77}

isolates the exponentiation part of the problem.

4

Determine the remainder of

7^{77}

divided by 19.

Using modulo operations,

7^1 = 7

,

7^2 = 49 \equiv 11

,

7^3 = 7 \times 11 = 77 \equiv 1 \pmod{19}

. Since

7^3 \equiv 1

, the cyclicity is 3. We divide the exponent by 3:

77 = 3 \times 25 + 2

. Thus,

7^{77} \equiv 7^2 \equiv 11 \pmod{19}

.

Finding the cycle of powers (where the remainder is 1) drastically reduces large exponents.

5

Multiply by the coefficient and convert to a positive remainder.

-2 \times 11 = -22

. To find the positive remainder, add multiples of 19:

-22 + 19 \times 2 = -22 + 38 = 16

.

Remainders must be positive integers strictly less than the divisor.

Key Concept

Modular Arithmetic and Fermat's Little Theorem

Alternative Method

Instead of finding the specific cyclicity of 3, Fermat's Little Theorem states that

a^{18} \equiv 1 \pmod{19}

. Dividing 77 by 18 gives a remainder of 5. Calculating

7^5 \pmod{19}

will also yield 11, leading to the same final answer.

Estimated Time:1m 30s

Question 220Question →

If the integer generated by the mathematical expression $15 + 35 \times 36^{87}$ is divided by $37$ , which of the following represents the correct positive remainder?

-20

17

24

13

Show answer & explanation

Answer: 17

Answer

17

Applying modular arithmetic properties, we can evaluate each part of the expression relative to the divisor

37

. First, the base

36

is equivalent to

-1 \pmod{37}

. Raising

-1

to the odd power of

87

results in

-1

. Substituting this back into the equation yields

15 + 35 \times (-1)

. Following the standard order of operations, we multiply first to get

-35

, and then add

15

, resulting in an intermediate value of

-20

. Since remainders must be positive, adding the divisor (

37

) to

-20

produces the final valid remainder of

17

.

Step-by-Step Solution

1

Simplify the base

36

modulo

37

.

36 \equiv -1 \pmod{37}

Reducing the base to a smaller equivalent modulus makes the large exponentiation manageable.

2

Evaluate the exponentiated term

36^{87} \pmod{37}

.

(-1)^{87} = -1

An odd power of a negative base retains the negative sign.

3

Substitute the reduced term back into the expression and apply the correct order of operations (BODMAS/PEMDAS).

15 + 35 \times (-1) = 15 - 35 = -20

Multiplication must be resolved prior to addition.

4

Convert the intermediate negative remainder to a standard positive remainder.

-20 + 37 = 17

A final remainder must always be a positive integer strictly less than the divisor.

Key Concept

Modular Arithmetic with Negative Bases and Operational Order

Estimated Time:1m 15s

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