All practice questions

Question 161Question →

A maritime patrol boat departs from its home harbor to secure coastal waters. It first sails $24\text{ km}$ strictly towards the West to investigate a radar blip. Finding nothing, it turns North and travels $7\text{ km}$ to a reported location. It then changes course, sailing $45\text{ km}$ East to follow a distress signal. Finally, it turns South and travels $27\text{ km}$ to successfully intercept the target. Calculate the exact straight-line distance from the home harbor to the interception point.

Show answer & explanation

Answer: 29

Answer

29

By resolving the boat's path into net vertical and horizontal components, we find its final position is

21\text{ km}

East and

20\text{ km}

South of its starting point. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), the shortest straight-line distance is the hypotenuse:

\sqrt{21^2 + 20^2} = 29\text{ km}

.

Step-by-Step Solution

1

Calculate the net horizontal (East-West) displacement.

21\text{ km}

East

The boat initially travels

24\text{ km}

West, then later travels

45\text{ km}

East. The net horizontal movement is

45 - 24 = 21\text{ km}

in the East direction.

2

Calculate the net vertical (North-South) displacement.

20\text{ km}

South

The boat travels

7\text{ km}

North, and later travels

27\text{ km}

South. The net vertical movement is

27 - 7 = 20\text{ km}

in the South direction.

3

Apply the Pythagorean theorem to find the shortest straight-line distance.

\sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29\text{ km}

The net East and South displacements form the two perpendicular legs of a right-angled triangle relative to the starting point. The straight-line distance is the hypotenuse.

Key Concept

Vector displacement across cardinal directions and Pythagorean theorem calculation.

Question 162Question →

On a straight railway track, a sequence of freight carriages is standing in a single line. Carriage $X$ is the $38^{\text{th}}$ carriage from the front end, and Carriage $Y$ is the $47^{\text{th}}$ carriage from the rear end. Carriage $Z$ is positioned exactly halfway between Carriage $X$ and Carriage $Y$ . If Carriage $Z$ is the $62^{\text{nd}}$ carriage from the rear end, what is the total number of freight carriages on the track?

Show answer & explanation

Answer: 114

Answer

114

The positional distance between Carriage

Y

and Carriage

Z

from the rear is 15. Since

Z

is the midpoint, Carriage

X

is another 15 positions ahead, making

X

the

77^{\text{th}}

carriage from the rear. Since

X

is also

38^{\text{th}}

from the front, the total number of carriages is calculated as

38 + 77 - 1 = 114

.

Step-by-Step Solution

1

Calculate the positional difference between Carriage

Y

and Carriage

Z

from the rear.

62 - 47 = 15

positions.

To find the distance from one end of the segment to its exact midpoint.

2

Determine Carriage

X

's position from the rear.

Carriage

X

is the

77^{\text{th}}

carriage from the rear.

Since

Z

is exactly halfway between

X

and

Y

, the positional difference between

Z

and

X

must also be 15. Thus,

62 + 15 = 77

.

3

Calculate the total number of carriages.

38 + 77 - 1 = 114

carriages.

The total number of items in a row equals (Position from front) + (Position from rear) - 1.

Key Concept

Calculating total count using dual positional ranks and midpoint symmetry in a sequence.

Question 163Question →

Evaluate the exact positive remainder that results from dividing $5^{75} - 45$ by $126$ .

Show answer & explanation

Answer: 80

Answer

The positive remainder is 80.

The exact remainder is determined using modular arithmetic rules. By noting that

5^3 = 125

, we see that

125 \equiv -1 \pmod{126}

. The expression

5^{75}

can be rewritten as

(5^3)^{25}

, which evaluates to

(-1)^{25} = -1

. Subtracting the constant

45

yields an intermediate remainder of

-46

. Because standard division requires a non-negative remainder, we must add the divisor (

126

) to

-46

, which results in

80

.

Step-by-Step Solution

1

Relate the base of the exponential term to the divisor

5^3 = 125 \equiv -1 \pmod{126}

Finding a power of the base that equals 1 or -1 modulo the divisor significantly simplifies calculations with large exponents.

2

Express the given exponent in terms of the chosen power

5^{75} = (5^3)^{25}

This allows for direct substitution of the modular equivalent.

3

Evaluate the remainder of the exponential portion

(-1)^{25} = -1 \pmod{126}

Substituting

-1

for

125

makes evaluating the

25^{th}

power trivial.

4

Incorporate the constant term into the expression

-1 - 45 = -46 \pmod{126}

The full expression is

5^{75} - 45

, so the constant must be subtracted from the remainder of the exponential part.

5

Convert the negative remainder into a standard positive remainder

-46 + 126 = 80

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

Key Concept

Modular Exponentiation and Negative Remainders

Alternative Method

You could apply the binomial theorem by rewriting

5^{75}

as

125^{25} = (126 - 1)^{25}

. Expanding this yields a series of terms that are all multiples of

126

, plus a final term of

(-1)^{25}

. This clearly leaves a remainder of

-1

. Subtracting

45

gives

-46

, which is then converted to

80

by adding

126

.

Estimated Time:1m 0s

Question 164Question →

If $P$ is the smallest prime number and $C$ is the smallest positive composite number, what is the value of $P + C$ ?

Show answer & explanation

Answer: 6

Answer

6

The smallest prime number is 2, as it is the first positive integer greater than 1 with exactly two distinct divisors. The smallest positive composite number is 4, as it is the first positive integer with more than two distinct divisors (1, 2, and 4). The number 1 is excluded from both categories. Therefore, their sum is 2 + 4 = 6.

Step-by-Step Solution

1

Identify the smallest prime number.

P = 2

A prime number is a positive integer greater than 1 that has exactly two positive divisors. The smallest such number is 2.

2

Identify the smallest positive composite number.

C = 4

A composite number is a positive integer that has at least one divisor other than 1 and itself. The numbers 1, 2, and 3 are not composite. The number 4 has divisors 1, 2, and 4, making it the smallest positive composite number.

3

Calculate the sum of P and C.

2 + 4 = 6

The question asks for the sum of the two identified values.

Key Concept

Basic definitions and properties of prime and composite numbers

Question 165Question →

What is the positive remainder when the product $54 \times 82$ is divided by $11$ ?

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Answer: 6

Answer

6

By finding the individual remainders of the factors (

-1

and

5

) and multiplying them, we get

-5

. Adding the divisor (

11

) to this negative remainder yields the correct positive remainder of

6

.

Step-by-Step Solution

1

Find the remainder of

54

when divided by

11

.

Since

11 \times 5 = 55

, we can write

54

as

55 - 1

. Thus,

54 \equiv -1 \pmod{11}

.

Using a negative remainder simplifies the subsequent multiplication.

2

Find the remainder of

82

when divided by

11

.

Since

11 \times 7 = 77

, we have

82 - 77 = 5

. Thus,

82 \equiv 5 \pmod{11}

.

We need the remainder of the second factor to apply the remainder theorem for products.

3

Multiply the individual remainders.

(-1) \times 5 = -5

According to modular arithmetic properties, the remainder of a product is the product of the individual remainders.

4

Convert the negative remainder to a positive remainder.

-5 + 11 = 6

The question specifically asks for the positive remainder. Adding the divisor to a negative remainder gives the equivalent positive remainder.

Key Concept

Remainder Theorem and Modular Arithmetic

Question 166Question →

Consider the three fractions $\frac{12}{5}$ , $\frac{24}{7}$ , and $\frac{36}{11}$ . What is the exact value of their Least Common Multiple (LCM)?

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Answer: 72

Answer

The Least Common Multiple (LCM) of the given fractions is 72.

To find the Least Common Multiple (LCM) of a set of fractions, we must divide the LCM of their numerators by the HCF of their denominators. The numerators (

12

,

24

, and

36

) have an LCM of

72

. The denominators (

5

,

7

, and

11

) share no common factors other than

1

, making their HCF

1

. Thus, the final LCM is

72 / 1 = 72

.

Step-by-Step Solution

1

State the standard formula for finding the LCM of multiple fractions.

LCM = (LCM of numerators) / (HCF of denominators).

This mathematical property governs how multiples apply to rational numbers.

2

Calculate the LCM of the numerators.

The LCM of

12

,

24

, and

36

is

72

.

72

is the smallest integer perfectly divisible by all three numerators.

3

Calculate the HCF of the denominators.

The HCF of

5

,

7

, and

11

is

1

.

The numbers

5

,

7

, and

11

are prime and share no common factors other than

1

.

4

Compute the final fraction LCM.

72 \div 1 = 72

.

Dividing the computed numerator LCM by the denominator HCF yields the answer.

Key Concept

Calculating the LCM of fractions using the specific formula relating numerators and denominators.

Question 167Question →

If the 9-digit number $94x257y36$ is exactly divisible by $88$ , what is the maximum possible value of the mathematical expression $(5x - 3y)$ ?

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Answer: 27

Answer

27

By applying the divisibility rules of 8 and 11, we isolate a finite set of possible digit combinations for x and y. Evaluating

(5x - 3y)

across all mathematically valid pairs proves that the pair

x=6, y=1

produces the highest numerical output of 27.

Step-by-Step Solution

1

Determine the co-prime factors of the divisor.

The number must be divisible by 8 and 11, since 88 = 8 * 11 and GCD(8, 11) = 1.

Applying individual divisibility rules for co-prime factors is the standard method for composite divisors.

2

Apply the divisibility rule for 8 to the last three digits (

y36

).

The possible values for

y

are 1, 3, 5, 7, and 9.

A number is divisible by 8 if its last three digits form a multiple of 8. 136, 336, 536, 736, and 936 all yield a remainder of 0 when divided by 8.

3

Apply the divisibility rule for 11 to the entire number.

The expression

(x + y + 4)

must be a multiple of 11.

The rule for 11 requires the difference between the alternating sum of digits to be divisible by 11. Odd positions sum to

(20 + x + y)

and even positions sum to

16

. Difference =

(x + y + 4)

.

4

Find all valid

(x, y)

coordinate pairs.

The valid digit pairs are (6, 1), (4, 3), (2, 5), (0, 7), and (9, 9).

Since

x

and

y

are single digits between 0 and 9,

(x + y + 4)

can only equal 11 or 22. This implies

x + y = 7

or

x + y = 18

.

5

Evaluate the target expression

(5x - 3y)

for all valid pairs.

The values obtained are 27, 11, -5, -21, and 18 respectively. The absolute maximum is 27.

Testing all valid edge cases guarantees that the maximum possible value is correctly identified without assumptions.

Key Concept

Solving for unknown digits using combined co-prime divisibility rules and optimizing a linear algebraic expression based on finite digit constraints.

Estimated Time:2m 30s

Question 168Question →

Determine the exact decimal value of the Least Common Multiple (LCM) for the two fractions $\frac{3}{5}$ and $\frac{9}{10}$ .

Show answer & explanation

Answer: 1.8

Answer

1.8

To find the LCM of fractions, we use the standard formula: LCM(numerators) / HCF(denominators). The numerators are 3 and 9, and their LCM is 9. The denominators are 5 and 10, and their HCF is 5. Dividing the LCM of the numerators by the HCF of the denominators gives 9/5. Converting this fraction to a decimal yields exactly 1.8.

Step-by-Step Solution

1

Identify the correct formula for finding the LCM of fractions.

LCM of fractions = LCM(numerators) / HCF(denominators)

This is the fundamental rule for calculating the least common multiple when dealing with rational numbers.

2

Calculate the Least Common Multiple of the numerators.

The numerators are 3 and 9. Their LCM is 9.

9 is the smallest integer that is perfectly divisible by both 3 and 9.

3

Calculate the Highest Common Factor of the denominators.

The denominators are 5 and 10. Their HCF is 5.

5 is the largest integer that perfectly divides both 5 and 10 without leaving a remainder.

4

Apply the formula and convert the resulting fraction to a decimal.

LCM = 9 / 5 = 1.8

The question specifically asks for the exact decimal value, so the fraction 9/5 must be converted to 1.8.

Key Concept

The Least Common Multiple (LCM) of a set of fractions is found by dividing the LCM of their numerators by the HCF of their denominators.

Question 169Question →

The product of two positive integers is $2160$ and their Highest Common Factor (HCF) is $12$ . What is their Least Common Multiple (LCM)?

Show answer & explanation

Answer: 180

Answer

180

For any two positive integers, the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM) is always equal to the product of the numbers themselves. Given that the product is

2160

and the HCF is

12

, the LCM is calculated by rearranging the formula to

\text{LCM} = \frac{\text{Product}}{\text{HCF}}

. Substituting the values gives

2160 \div 12 = 180

.

Step-by-Step Solution

1

Identify the fundamental formula connecting HCF, LCM, and the product of two positive integers.

\text{HCF} \times \text{LCM} = \text{Product}

This relationship allows you to find one missing value when the other two are known.

2

Substitute the known values from the problem into the equation.

12 \times \text{LCM} = 2160

The problem explicitly states that the product is

2160

and the HCF is

12

.

3

Solve for the LCM by dividing both sides of the equation by

12

.

\text{LCM} = 180

Isolating the LCM variable provides the final requested value.

Key Concept

Relationship between the product of two numbers and their HCF and LCM

Question 170Question →

At exactly $4:28$ PM, what is the exact acute angle (in degrees) between the hour hand and the minute hand of a standard analog clock?

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Answer: 34

Answer

34

The angle is determined by finding the exact position of both hands relative to the 12 o'clock mark. The minute hand is at 168 degrees (28 minutes multiplied by 6 degrees). The hour hand is at 134 degrees, which includes its base position at 4 o'clock (120 degrees) plus the drift from the 28 minutes that have passed (14 degrees). The absolute difference between these two positions is exactly 34 degrees.

Step-by-Step Solution

1

Calculate the angle of the minute hand from the 12 o'clock position.

28 \times 6^\circ = 168^\circ

The minute hand moves at a rate of 6 degrees per minute.

2

Calculate the base angle of the hour hand exactly at 4 o'clock.

4 \times 30^\circ = 120^\circ

Each hour marking on the clock represents 30 degrees from the top.

3

Calculate the drift of the hour hand caused by the 28 elapsed minutes.

28 \times 0.5^\circ = 14^\circ

The hour hand continuously moves at a rate of 0.5 degrees per minute.

4

Determine the true position of the hour hand.

120^\circ + 14^\circ = 134^\circ

Adding the drift to the base position gives the exact current location of the hour hand.

5

Calculate the acute angle between the two hands.

|168^\circ - 134^\circ| = 34^\circ

The angle between the hands is the absolute difference between their positions from the 12 o'clock mark.

Key Concept

Calculating the precise angle between clock hands by accounting for hour-hand drift

Question 171Question →

A wildlife biologist tracking a radio-collared leopard starts at a research outpost and drives $10\text{ km}$ straight East. She then turns $45^{\circ}$ to her left and drives $10\sqrt{2}\text{ km}$ along a dirt trail. Next, she turns $90^{\circ}$ to her right and travels $4\sqrt{2}\text{ km}$ through the brush. Finally, she turns $135^{\circ}$ to her left and drives $1\text{ km}$ straight North to locate the leopard. What is the shortest straight-line distance, in kilometers, between the research outpost and the leopard's final location?

Show answer & explanation

Answer: 25

Answer

25

The correct answer is found by tracking the Cartesian coordinates after each vector movement. By breaking diagonal distances into their horizontal and vertical components, the final position is located exactly at

(24, 7)

. Using the distance formula

\sqrt{24^2 + 7^2}

, the shortest direct distance from the origin is

25\text{ km}

.

Step-by-Step Solution

1

Establish a coordinate system with the starting point at

(0,0)

and translate the first movement of

10\text{ km}

East.

Current position is

(10, 0)

.

East corresponds to the positive x-axis direction.

2

Determine the new heading after turning

45^{\circ}

left from East, and calculate the displacement for

10\sqrt{2}\text{ km}

.

New heading is North-East. Displacement is

(10, 10)

, making the current position

(20, 10)

.

A

45^{\circ}

left turn from East (

0^{\circ}

) results in

45^{\circ}

(NE). The vector components are

d \times \cos(45^{\circ})

and

d \times \sin(45^{\circ})

.

3

Determine the new heading after turning

90^{\circ}

right from North-East, and calculate the displacement for

4\sqrt{2}\text{ km}

.

New heading is South-East. Displacement is

(4, -4)

, making the current position

(24, 6)

.

A

90^{\circ}

right turn from NE (

45^{\circ}

) results in SE (

-45^{\circ}

). The vector components are

d \times \cos(-45^{\circ})

and

d \times \sin(-45^{\circ})

.

4

Determine the new heading after a

135^{\circ}

left turn from South-East, and calculate the displacement for

1\text{ km}

.

New heading is North. Displacement is

(0, 1)

, making the final position

(24, 7)

.

A

135^{\circ}

left turn (positive rotation) from SE (

-45^{\circ}

) yields

-45^{\circ} + 135^{\circ} = 90^{\circ}

(North).

5

Apply the Pythagorean theorem to find the straight-line distance from the origin

(0,0)

to the final coordinates

(24, 7)

.

The shortest distance is

25\text{ km}

.

The straight-line distance between two points on a Cartesian plane is given by

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

.

Key Concept

Vector displacement, angular turns mapping to cardinal/ordinal directions, and the Pythagorean theorem.

Question 172Question →

Consider the positive integer $2160$ . How many of its positive factors are perfect squares?

Show answer & explanation

Answer: 6

Answer

6

The prime factorization of

2160

is

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

. For a factor to be a perfect square, all the exponents in its prime factorization must be even integers. Analyzing the bases: the base

2

can have an exponent of

0, 2, \text{ or } 4

(three options). The base

3

can have an exponent of

0 \text{ or } 2

(two options). The base

5

can only have an exponent of

0

to remain even (one option). Multiplying the number of possibilities for each base gives

3 \times 2 \times 1 = 6

total perfect square factors.

Step-by-Step Solution

1

Find the prime factorization of

2160

.

2160 = 2^4 \times 3^3 \times 5^1

Prime factorization is necessary to analyze the properties of the number's factors.

2

Identify the mathematical condition for a factor to be a perfect square.

Any perfect square factor must have the form

2^{2x} \times 3^{2y} \times 5^{2z}

where the exponents are even.

Perfect squares require all prime factors to appear in pairs.

3

Count the number of valid even exponents for each prime factor.

For

2

: exponents

0, 2, 4

(

3

choices). For

3

: exponents

0, 2

(

2

choices). For

5

: exponent

0

(

1

choice).

The exponent of each prime in the factor cannot exceed its exponent in the original number.

4

Calculate the total number of perfect square factors.

3 \times 2 \times 1 = 6

The Fundamental Principle of Counting states we must multiply the independent choices to get the total number of combinations.

Key Concept

Using prime factorization to determine the properties and quantity of specific types of factors (perfect squares).

Question 173Question →

A multinational corporation allocated a total annual training budget of 140,000 dollars across its three major divisions: Sales, Information Technology (IT), and Human Resources (HR). According to the financial distribution guidelines, the IT division received exactly 20% more funding than the Sales division. Meanwhile, the HR division was allocated exactly half of the amount given to the IT division. If the HR division dedicates 40% of its allocated training budget specifically to onboarding programs, what is the exact amount (in dollars) spent by the HR division on onboarding?

Show answer & explanation

Answer: 12000

Answer

12000

By converting the text into an algebraic equation, we establish the Sales budget as

x

, IT as

1.2x

, and HR as

0.6x

. The sum

2.8x

equals 140,000 dollars, making

x = 50,000

. The HR division receives

0.6x

, which is 30,000 dollars. Finally, taking 40% of the HR division's 30,000 dollar budget yields 12,000 dollars for onboarding programs.

Step-by-Step Solution

1

Define algebraic variables for each division's allocation.

Sales =

x

, IT =

1.2x

, HR =

0.6x

To translate the relative proportional statements from the paragraph into a solvable mathematical model.

2

Set up an equation representing the total corporate training budget.

x + 1.2x + 0.6x = 140,000

The sum of the three divisional budgets must equal the total allocated amount.

3

Solve the equation for

x

to find the Sales division's baseline budget.

2.8x = 140,000 \Rightarrow x = 50,000

To isolate the base variable from which the other departmental budgets are derived.

4

Calculate the total training budget allocated specifically to the HR division.

HR =

0.6 \times 50,000 = 30,000

The question asks about a subset of the HR budget, so the total HR budget must be known first.

5

Calculate 40% of the HR division's budget.

0.40 \times 30,000 = 12,000

The paragraph specifies that exactly 40% of the HR training budget goes to onboarding programs.

Key Concept

Extracting quantitative relationships from text to form algebraic equations and calculating sequential percentages.

Question 174Question →

Consider the following $8$ mathematical expressions:

I. $\pi - \frac{22}{7}$
II. $\frac{\sqrt{27}}{\sqrt{3}}$
III. $(\sqrt{3} - \sqrt{2})^2$
IV. The infinite decimal $0.101101110\dots$ (where the number of consecutive $1$ s increases by one each time)
V. $1.414$
VI. $\sqrt{10} \times \sqrt{2.5}$
VII. $e^0$
VIII. $\sqrt{2} + \sqrt{8}$

How many of the above expressions evaluate to a rational number?

Show answer & explanation

Answer: 4

Answer

Exactly 4 of the expressions evaluate to a rational number.

Out of the 8 expressions provided, exactly 4 result in rational numbers: Expression II (simplifies to 3), Expression V (terminating decimal 1414/1000), Expression VI (simplifies to sqrt(25) = 5), and Expression VII (simplifies to 1). The remaining 4 expressions yield irrational results.

Step-by-Step Solution

1

Evaluate expression I (pi minus 22/7)

The expression evaluates to an irrational number.

The number pi is irrational, while 22/7 is a rational approximation. The difference between an irrational number and a rational number is always irrational.

2

Evaluate expression II (sqrt(27) / sqrt(3))

The expression simplifies to 3, which is a rational number.

Using the properties of radicals, sqrt(27) / sqrt(3) = sqrt(27/3) = sqrt(9) = 3.

3

Evaluate expression III (square of (sqrt(3) - sqrt(2)))

The expression expands to 5 - 2*sqrt(6), which is an irrational number.

Applying the binomial square formula (a-b)^2 = a^2 - 2ab + b^2 yields 3 - 2*sqrt(6) + 2. Since sqrt(6) is irrational, the entire expression is irrational.

4

Evaluate expression IV (the pattern decimal 0.101101110...)

The expression is an irrational number.

The decimal is non-terminating and non-periodic (the pattern changes constantly by adding an extra 1), which is the definition of an irrational decimal.

5

Evaluate expression V (1.414)

The expression is a rational number.

1.414 is a terminating decimal. Any terminating decimal can be written as a fraction of integers (1414/1000), making it rational.

6

Evaluate expression VI (sqrt(10) * sqrt(2.5))

The expression simplifies to 5, which is a rational number.

Multiplying the terms inside the radicals gives sqrt(10 * 2.5) = sqrt(25) = 5.

7

Evaluate expression VII (e^0)

The expression simplifies to 1, which is a rational number.

Any non-zero real number raised to the power of 0 equals 1.

8

Evaluate expression VIII (sqrt(2) + sqrt(8))

The expression simplifies to 3*sqrt(2), which is an irrational number.

sqrt(8) can be simplified to 2*sqrt(2). Adding sqrt(2) gives 3*sqrt(2), which remains an irrational product.

9

Count the total number of rational expressions

Expressions II, V, VI, and VII are rational. Total count is 4.

Identifying the rational outcomes from the previous steps.

Key Concept

Properties and Definitions of Rational and Irrational Numbers

Estimated Time:2m 30s

Question 175Question →

At exactly 10:10 AM, the hour hand of a standard analog clock is mechanically locked in its precise current position. The minute hand is then manually rotated forward (clockwise) by exactly $210^{\circ}$ . Calculate the new angle (in degrees) formed between the hour and minute hands after this adjustment. Provide the smaller of the two possible angles.

Show answer & explanation

Answer: 35

Answer

The smaller angle between the hands after the adjustment is 35 degrees.

The correct calculation properly tracks the absolute angular positions from the 12 o'clock reference, correctly adding the

5^{\circ}

proportional movement of the hour hand before finding the difference against the shifted minute hand.

Step-by-Step Solution

1

Find the initial position of the minute hand at 10:10 AM relative to the 12 o'clock mark.

The minute hand is at the 10-minute mark. Since each minute is

6^{\circ}

, its position is

10 \times 6^{\circ} = 60^{\circ}

.

Establishing the starting angle of the minute hand is required before applying the rotation.

2

Find the exact initial position of the hour hand at 10:10 AM.

The 10 o'clock mark is at

10 \times 30^{\circ} = 300^{\circ}

. In 10 minutes, the hour hand drifts

10 \times 0.5^{\circ} = 5^{\circ}

. The exact position is

300^{\circ} + 5^{\circ} = 305^{\circ}

.

The hour hand continuously moves. It does not stay exactly on the 10 while the minute hand moves.

3

Calculate the new position of the minute hand after the manual rotation.

The new minute hand position is

60^{\circ} + 210^{\circ} = 270^{\circ}

.

The problem states the minute hand is pushed forward by exactly

210^{\circ}

.

4

Calculate the angle between the locked hour hand and the new minute hand position.

The difference is

305^{\circ} - 270^{\circ} = 35^{\circ}

.

Subtracting the smaller angular position from the larger one gives the acute interior angle between the hands.

Key Concept

Clock hand angular positioning, rotational displacement, and proportional hour-hand drift.

Question 176Question →

Trees are planted in a single straight line along a newly constructed highway. An Oak tree is the $42^{\text{nd}}$ tree from the northern end of the line, and a Maple tree is the $58^{\text{th}}$ tree from the southern end. If there are exactly $18$ trees planted between the Oak tree and the Maple tree, what is the minimum possible total number of trees in this line?

Show answer & explanation

Answer: 80

Answer

80

To find the minimum possible number of trees, we must assume an overlapping scenario where the Maple tree is situated north of the Oak tree. In this arrangement, adding their positions from opposite ends (

42 + 58

) double-counts the

18

trees between them, as well as the Oak and Maple trees themselves. Therefore, the minimum total is

42 + 58 - (18 + 2) = 80

.

Step-by-Step Solution

1

Identify the given positional values and the condition.

Oak tree = 42nd from North, Maple tree = 58th from South, trees between = 18.

These are the core variables needed to determine the total number of items in the line.

2

Determine the spatial arrangement required for a 'minimum' total.

The two trees must overlap. The Maple tree (counted from the South) must be located further North than the Oak tree (counted from the North).

If the positions do not overlap, we get the maximum possible number of trees. Overlapping them minimizes the total line length.

3

Apply the overlapping rank formula.

Total = (Position from one end) + (Position from other end) - (Number in between) - 2.

Adding the two ranks double-counts the items between them AND double-counts the two specific boundary items (the Oak and Maple trees) themselves.

4

Substitute the values and calculate the result.

Total = 42 + 58 - 18 - 2 = 100 - 20 = 80.

Computing the arithmetic expression gives the minimum number of trees.

Key Concept

Calculating minimum capacity in overlapping queues

Question 177Question →

In an advanced error-checking algorithm, a data packet's integrity is verified by calculating a specific numerical signature. This signature is defined as the remainder when the value $V = 112^{2029} - 122^{2026}$ is divided by $13$ . What is the numerical signature of this data packet?

Show answer & explanation

Answer: 9

Answer

The numerical signature of the data packet is 9.

The expression evaluates to

-4 \pmod{13}

after simplifying the bases and exponents. Converting this to a valid positive remainder by adding the modulus

13

yields the correct answer,

9

.

Step-by-Step Solution

1

Reduce the base 112 modulo 13.

112 = 13 \times 8 + 8

, so

112 \equiv 8 \pmod{13}

.

In modular arithmetic, bases can be replaced by their remainders to simplify exponentiation.

2

Reduce the base 122 modulo 13.

122 = 13 \times 9 + 5

, so

122 \equiv 5 \pmod{13}

.

This simplifies the second term of the expression.

3

Apply Fermat's Little Theorem to simplify the exponent for the first term.

Since 13 is prime,

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

. For

8^{2029}

, dividing

2029

by

12

leaves a remainder of

1

(

2029 = 12 \times 169 + 1

). Thus,

8^{2029} \equiv 8^1 = 8 \pmod{13}

.

Large powers can be reduced by finding the remainder of the exponent divided by

p-1

.

4

Apply Fermat's Little Theorem to simplify the exponent for the second term.

For

5^{2026}

, dividing

2026

by

12

leaves a remainder of

10

(

2026 = 12 \times 168 + 10

). Thus,

5^{2026} \equiv 5^{10} \pmod{13}

.

This reduces the exponent to a manageable size.

5

Evaluate

5^{10} \pmod{13}

.

Since

5^2 = 25 \equiv -1 \pmod{13}

, we calculate

5^{10} = (5^2)^5 \equiv (-1)^5 = -1 \equiv 12 \pmod{13}

.

Using negative remainders for intermediate steps significantly speeds up manual exponentiation.

6

Subtract the evaluated terms and find the final positive remainder.

8 - 12 = -4

. To find the positive remainder, add the modulus 13:

-4 + 13 = 9

.

The final remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Modular arithmetic, Fermat's Little Theorem, and handling negative remainders.

Alternative Method

Instead of using Fermat's Little Theorem (mod 12), you can find the exact order of the bases. Notice that

8^2 = 64 \equiv -1 \pmod{13}

, so

8^4 \equiv 1 \pmod{13}

. Similarly,

5^2 = 25 \equiv -1 \pmod{13}

, so

5^4 \equiv 1 \pmod{13}

. Both bases have an order of 4 modulo 13. Dividing the exponents by 4:

2029 \div 4

leaves a remainder of 1, so

8^{2029} \equiv 8^1 = 8

.

2026 \div 4

leaves a remainder of 2, so

5^{2026} \equiv 5^2 = 25 \equiv -1 \equiv 12

. The final subtraction is

8 - 12 = -4 \equiv 9 \pmod{13}

. This method requires less calculation.

Estimated Time:2m 0s

Question 178Question →

What is the total number of positive divisors of the integer $10800$ that are perfect squares?

Show answer & explanation

Answer: 12

Answer

12

The correct answer is found by prime factorizing

10800

into

2^4 \times 3^3 \times 5^2

. A factor is a perfect square only when its prime exponents are all even integers. Counting the available even exponents from 0 up to the maximum exponent for each prime gives 3 choices for base 2 (0, 2, 4), 2 choices for base 3 (0, 2), and 2 choices for base 5 (0, 2). Multiplying these independent choices gives

3 \times 2 \times 2 = 12

.

Step-by-Step Solution

1

Find the prime factorization of

10800

.

10800 = 108 \times 100 = (2^2 \times 3^3) \times (2^2 \times 5^2) = 2^4 \times 3^3 \times 5^2

The prime factorization is necessary to systematically identify and count the divisors of the number based on exponent combinations.

2

Identify the mathematical condition for a divisor to be a perfect square.

A divisor

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

is a perfect square if and only if the exponents

a

,

b

, and

c

are all even integers.

Perfect squares must have even exponents in their prime factorization so that their square roots are integers.

3

List the valid even exponent values for each prime base.

For base 2 (max exponent 4), valid even values are 0, 2, 4 (3 options). For base 3 (max exponent 3), valid even values are 0, 2 (2 options). For base 5 (max exponent 2), valid even values are 0, 2 (2 options).

These are the exponents that keep the divisor as a perfect square while still remaining a factor of

10800

.

4

Calculate the total number of perfect square divisors.

3 \times 2 \times 2 = 12

Applying the fundamental counting principle, we multiply the number of independent choices for each prime exponent.

Key Concept

Identifying perfect square factors using prime factorization

Estimated Time:1m 30s

Question 179Question →

A solid wooden cube is painted Black on two adjacent faces, while the remaining four faces are painted White. The large cube is then cut into $64$ identical smaller cubes. What is the total number of smaller cubes that have at least one Black face?

Show answer & explanation

Answer: 28

Answer

28

The large cube is divided into

64

smaller cubes, which means it forms a

4 \times 4 \times 4

grid. A single face of this large cube contains

4 \times 4 = 16

smaller cubes. Since two adjacent faces are painted Black, we count the cubes on both faces. The first face has

16

Black cubes. The second face also has

16

Black cubes, but it shares one edge with the first face. This shared edge consists of

4

smaller cubes. If we simply add

16 + 16 = 32

, we double-count the

4

cubes on the shared edge. Subtracting the overlap gives

32 - 4 = 28

unique smaller cubes that have at least one Black face.

Step-by-Step Solution

1

Find the edge length of the large cube in terms of smaller cubes.

n = 4

The total number of smaller cubes is

64

, and the total volume of a cube is

n^3

. Therefore,

n = \sqrt[3]{64} = 4

.

2

Count the smaller cubes on the first Black face.

16

cubes

Each face of an

n \times n \times n

cube contains

n^2

smaller cubes. Here,

4^2 = 16

cubes.

3

Count the smaller cubes on the second Black face and identify the overlap.

The second face adds

16

cubes, but the two faces share an edge of

4

cubes.

Since the two Black faces are adjacent, they intersect along one edge. The cubes on this edge are counted as part of both faces.

4

Calculate the total unique cubes with at least one Black face.

28

cubes

Using the inclusion-exclusion principle:

16

(first face) +

16

(second face) -

4

(shared edge) =

28

.

Key Concept

Calculation of painted smaller cubes from a larger cut cube, specifically handling adjacent face intersections.

Question 180Question →

An industrial control system tracks part serial numbers using a cyclic validation algorithm. A batch identifier $N$ is generated using the formula $N = 65^{202} - 32^{145} - 45$ . The system validates this batch by computing the positive remainder when $N$ is divided by the base modulus $33$ . What is the expected positive remainder for this batch?

Show answer & explanation

Answer: 23

Answer

The expected positive remainder is 23.

By applying the principles of modular arithmetic, we can simplify each term of the expression

65^{202} - 32^{145} - 45

modulo

33

. First,

65 \equiv -1 \pmod{33}

, so

65^{202} \equiv (-1)^{202} = 1

. Next,

32 \equiv -1 \pmod{33}

, so

32^{145} \equiv (-1)^{145} = -1

. Substituting these simplified values back into the expression yields

1 - (-1) - 45 = 2 - 45 = -43

. To find the positive remainder when dividing by

33

, we add a multiple of

33

(in this case,

66

) to

-43

, which results in

23

.

Step-by-Step Solution

1

Express the base numbers

65

and

32

modulo

33

.

65 \equiv -1 \pmod{33}

and

32 \equiv -1 \pmod{33}

Finding remainders close to

1

or

-1

significantly simplifies the evaluation of large exponents.

2

Evaluate the first term,

65^{202}

modulo

33

.

(-1)^{202} = 1

An even power of a negative one results in positive one.

3

Evaluate the second term,

32^{145}

modulo

33

.

(-1)^{145} = -1

An odd power of a negative one results in negative one.

4

Substitute the simplified terms back into the original expression modulo

33

.

N \equiv 1 - (-1) - 45 \equiv 2 - 45 \equiv -43 \pmod{33}

Combines all parts of the expression to find the overall remainder before final normalization.

5

Convert the negative remainder into a valid positive remainder.

-43 \equiv -43 + 66 = 23 \pmod{33}

Remainders must be positive integers between

0

and the modulus minus one. Adding the smallest multiple of

33

that is greater than

43

(which is

66

) gives the valid positive remainder.

Key Concept

Modular Arithmetic and Remainder Theorem with Negative Remainders

Alternative Method

You could also simplify the constant term first:

-45 \equiv -12 \equiv 21 \pmod{33}

. Then the expression becomes

1 - (-1) + 21 = 23

.

Estimated Time:1m 30s

Answer

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