All practice questions

If $N = 3^{12} - 1$ , what is the sum of all the distinct prime factors of $N$ ?

Show answer & explanation

Answer: 100

Answer

The sum of all the distinct prime factors of

N

is 100.

Factoring

3^{12} - 1

via difference of squares and sum of cubes yields

(3^6 - 1)(3^6 + 1) = (2^3 \times 7 \times 13)(2 \times 5 \times 73) = 2^4 \times 5 \times 7 \times 13 \times 73

. The distinct prime factors are 2, 5, 7, 13, and 73, which sum to 100.

Step-by-Step Solution

1

Decompose

N = 3^{12} - 1

using the difference of squares identity

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

.

N = (3^6 - 1)(3^6 + 1)

Breaking down large powers of integers into products of smaller terms allows for systematic prime factor determination.

2

Completely factor the term

(3^6 - 1)

.

3^6 - 1 = (3^3 - 1)(3^3 + 1) = 26 \times 28 = (2 \times 13) \times (2^2 \times 7) = 2^3 \times 7 \times 13

Applying the difference of squares identity iteratively converts the term into small arithmetic integers with obvious prime factorizations.

3

Completely factor the term

(3^6 + 1)

using the sum of cubes identity

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

.

3^6 + 1 = (3^2)^3 + 1 = (3^2 + 1)((3^2)^2 - 3^2 + 1) = 10 \times 73 = 2 \times 5 \times 73

The factor 73 is prime because it is not divisible by any prime numbers less than or equal to

\sqrt{73} \approx 8.54

(namely 2, 3, 5, and 7).

4

Combine the component prime factorizations to list all distinct prime factors of

N

.

N = 2^4 \times 5 \times 7 \times 13 \times 73

, so the set of distinct prime factors is \{2, 5, 7, 13, 73\}.

Each prime base is included exactly once regardless of its exponent.

5

Sum the distinct prime factors.

2 + 5 + 7 + 13 + 73 = 100

Adding the unique prime factors yields the requested value.

Key Concept

Prime Factorization using Algebraic Polynomial Identities

Estimated Time:2m 30s

Question 62Question →

A positive integer $n$ has a prime factorization of the form $2^a \cdot 3^b \cdot 5^c$ , where $a$ , $b$ , and $c$ are non-negative integers. If $n$ is a multiple of $20$ but not a multiple of $40$ , is not divisible by $9$ , and has exactly $12$ positive integer divisors, what is the sum of all possible values of $n$ ?

Show answer & explanation

Answer: 560

Answer

The sum of all possible values of

n

is 560.

To determine the sum of all possible values of

n

, first analyze the given conditions on the prime exponents of

n = 2^a \cdot 3^b \cdot 5^c

. Divisibility by

20 = 2^2 \cdot 5^1

requires

a \ge 2

and

c \ge 1

. Since

n

is not divisible by

40 = 2^3 \cdot 5^1

, the exponent

a

must equal 2. Since

n

is not divisible by

9 = 3^2

, the exponent

b

must be strictly less than 2, meaning

b

can be 0 or 1. The total number of positive integer divisors is given by

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

. Substituting

a = 2

gives

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

, or

(b+1)(c+1) = 4

. If

b = 0

, then

c+1 = 4 \implies c = 3

, which gives

n = 2^2 \cdot 3^0 \cdot 5^3 = 500

. If

b = 1

, then

c+1 = 2 \implies c = 1

, which gives

n = 2^2 \cdot 3^1 \cdot 5^1 = 60

. The sum of these two valid values is

500 + 60 = 560

.

Step-by-Step Solution

1

Analyze the divisibility constraints to find values/ranges for exponents

a

,

b

, and

c

.

a = 2

,

c \ge 1

, and

b \in \{0, 1\}

.

Divisibility by 20 (

2^2 \cdot 5^1

) requires

a \ge 2

and

c \ge 1

. Non-divisibility by 40 (

2^3 \cdot 5^1

) restricts

a < 3

, so

a = 2

. Non-divisibility by 9 (

3^2

) restricts

b < 2

.

2

Apply the divisor count formula

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

.

(b+1)(c+1) = 4

.

Plugging

a = 2

into

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

yields

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

, which simplifies to

(b+1)(c+1) = 4

.

3

Evaluate the cases for

b = 0

and

b = 1

.

The valid values for

n

are

500

and

60

.

When

b = 0

,

c+1 = 4 \implies c = 3

, yielding

n = 2^2 \cdot 3^0 \cdot 5^3 = 500

. When

b = 1

,

c+1 = 2 \implies c = 1

, yielding

n = 2^2 \cdot 3^1 \cdot 5^1 = 60

.

4

Calculate the sum of all valid integers

n

.

560

500 + 60 = 560

.

Key Concept

Divisor Count Formula and Divisibility Constraints

Question 63Question →

An interior designer is arranging a row of 7 decorative wall tiles consisting of 3 identical blue tiles, 2 identical yellow tiles, and 2 identical red tiles. If the 2 red tiles cannot be placed next to each other, how many distinct arrangements of the 7 tiles are possible?

Show answer & explanation

Answer: 150

Answer

The total number of distinct arrangements possible is 150.

To find the total number of distinct arrangements where no two red tiles are adjacent, first calculate the arrangements of the 5 non-restricted tiles (3 blue, 2 yellow), which is

\frac{5!}{3!2!} = 10

. Placing 5 tiles creates 6 available spaces (including the two ends). Selecting 2 of these 6 spaces for the 2 identical red tiles yields

\binom{6}{2} = 15

choices. By the Fundamental Counting Principle, the total number of valid arrangements is

10 \times 15 = 150

.

Step-by-Step Solution

1

Calculate the number of distinct ways to arrange the non-restricted tiles (3 identical blue and 2 identical yellow).

The number of distinct arrangements of the 5 non-red tiles is

\frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10

.

Arranging all non-restricted tiles first creates the specific positions into which the restricted tiles can be inserted.

2

Determine the number of valid positions for the 2 identical red tiles such that no two are adjacent.

Placing 5 tiles creates 6 available insertion slots (one at each end and four between tiles). Choosing 2 distinct slots out of 6 gives

\binom{6}{2} = \frac{6 \times 5}{2} = 15

ways.

Selecting 2 distinct slots ensures that every selected space holds at most one red tile, guaranteeing that no two red tiles are adjacent.

3

Multiply the results from Step 1 and Step 2 using the Fundamental Counting Principle.

10 \times 15 = 150

.

Each arrangement of non-red tiles can be independently paired with any valid placement of the red tiles.

Key Concept

Counting arrangements of identical items with non-adjacency restrictions using the slotting method.

Estimated Time:1m 30s

Question 64Question →

If $x$ is a negative integer and $y$ is a positive integer, what is the value of $\frac{|x - y| - |y - x|}{x + y + 100}$ ?

Show answer & explanation

Answer: 0

Answer

0

Given

x < 0

and

y > 0

, the quantity

x - y

is strictly negative, which means

|x - y| = -(x - y) = y - x

. The quantity

y - x

is strictly positive, so

|y - x| = y - x

. Thus, the numerator simplifies to

(y - x) - (y - x) = 0

. Dividing zero by any non-zero denominator yields

0

.

Step-by-Step Solution

1

Determine the signs of the terms inside the absolute value functions

Since

x < 0

and

y > 0

,

x - y < 0

and

y - x > 0

.

Subtracting a positive number from a negative number yields a negative value, while subtracting a negative number from a positive number yields a positive value.

2

Simplify the absolute value expressions

|x - y| = y - x

and

|y - x| = y - x

.

The absolute value of a negative number is its negation, and the absolute value of a positive number is the number itself.

3

Compute the difference in the numerator

|x - y| - |y - x| = (y - x) - (y - x) = 0

.

Subtracting an algebraic expression from an identical expression results in zero.

4

Evaluate the entire fraction

\frac{0}{x + y + 100} = 0

.

Zero divided by any non-zero real number is zero.

Key Concept

Positive and Negative Number Properties with Absolute Value

Question 65Question →

At the beginning of a fiscal year, a municipal transit authority allocated its capital expenditure budget among three projects: Bus Rapid Transit, Rail Modernization, and Station Upgrades. Exactly $0.30$ of the total budget was allocated to Bus Rapid Transit. Of the remaining budget, exactly $\frac{3}{7}$ was allocated to Rail Modernization, and the rest was allocated to Station Upgrades. By the end of the year, expenditures on Bus Rapid Transit exceeded its initial allocation by $25\%$ , expenditures on Rail Modernization were $15\%$ below its initial allocation, and expenditures on Station Upgrades exceeded its initial allocation by $10\%$ . By what percent did the transit authority's total expenditures across all three projects exceed its initial total budget?

Show answer & explanation

Answer: 7

Answer

The total expenditures across all three projects exceeded the initial total budget by

7\%

.

To solve this problem, represent the total initial budget as

B

. The Bus Rapid Transit allocation is

0.30B

, leaving

0.70B

. Rail Modernization receives

\frac{3}{7}

of

0.70B

, which equals

0.30B

. The remaining portion for Station Upgrades is

0.70B - 0.30B = 0.40B

. End-of-year expenditures are calculated by multiplying each allocation by its respective growth multiplier: Bus Rapid Transit is

0.30B \times 1.25 = 0.375B

, Rail Modernization is

0.30B \times 0.85 = 0.255B

, and Station Upgrades is

0.40B \times 1.10 = 0.44B

. Summing these expenditures gives

0.375B + 0.255B + 0.44B = 1.07B

. Comparing

1.07B

to the initial

1.00B

reveals an overall increase of

0.07B

, or

7\%

.

Step-by-Step Solution

1

Express the initial allocations for each project as fractions of the total budget B

Bus Rapid Transit =

0.30B

, Rail Modernization =

0.30B

, Station Upgrades =

0.40B

Bus Rapid Transit is explicitly

0.30B

. The remaining

0.70B

is split such that Rail Modernization receives

\frac{3}{7} \times 0.70B = 0.30B

, leaving

0.70B - 0.30B = 0.40B

for Station Upgrades.

2

Apply the individual percentage changes to determine end-of-year expenditures

Bus Rapid Transit =

0.375B

, Rail Modernization =

0.255B

, Station Upgrades =

0.44B

A

25\%

increase corresponds to a multiplier of

1.25

, a

15\%

decrease corresponds to a multiplier of

0.85

, and a

10\%

increase corresponds to a multiplier of

1.10

.

3

Sum the project expenditures and calculate the net percent change relative to B

Total expenditure =

1.07B

, corresponding to a

7\%

net increase

Adding

0.375B + 0.255B + 0.44B

yields

1.07B

. Subtracting the original budget

1.00B

gives

0.07B

, which is

7\%

of

B

.

Key Concept

Weighted Percentage Changes and Sequential Fraction-Decimal Operations

Question 66Question →

If $x$ is a negative real number and $y$ is a positive real number, what is the value of $\frac{x}{|x|} + \frac{|y|}{y}$ ?

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Answer: 0

Answer

The value of the expression is 0.

For any negative number

x

, the ratio

\frac{x}{|x|}

evaluates to

-1

because

|x| = -x

. For any positive number

y

, the ratio

\frac{|y|}{y}

evaluates to

1

because

|y| = y

. Summing

-1

and

1

results in

0

.

Step-by-Step Solution

1

Evaluate the first term for a negative variable

-1

By definition of absolute value, if

x < 0

, then

|x| = -x

, making

\frac{x}{|x|} = \frac{x}{-x} = -1

.

2

Evaluate the second term for a positive variable

1

If

y > 0

, then

|y| = y

, making

\frac{|y|}{y} = \frac{y}{y} = 1

.

3

Sum the simplified values

0

Combining

-1

and

1

yields

-1 + 1 = 0

.

Key Concept

Properties of Absolute Value and Signs of Numbers

Question 67Question →

If $k$ is a positive integer such that $\sqrt{8^{10} + 2^{23} + 4^7} = 2^k + 128$ , what is the value of $k$ ?

Show answer & explanation

Answer: 15

Answer

The value of

k

is 15.

Converting all terms under the square root to base 2 produces

\sqrt{2^{30} + 2^{23} + 2^{14}}

. Recognizing that

(2^{15} + 2^7)^2 = (2^{15})^2 + 2(2^{15})(2^7) + (2^7)^2 = 2^{30} + 2^{23} + 2^{14}

, taking the square root yields

2^{15} + 2^7 = 2^{15} + 128

. Matching this with

2^k + 128

yields

k = 15

.

Step-by-Step Solution

1

Convert terms under the square root to base 2.

The radical expression becomes

\sqrt{2^{30} + 2^{23} + 2^{14}}

.

Expressing terms with the same base allows exponent rules and algebraic identity recognition.

2

Identify the expression under the radical as a perfect square of the form

(a + b)^2

.

Setting

a = 2^{15}

and

b = 2^7

gives

2ab = 2 \cdot 2^{15} \cdot 2^7 = 2^{23}

, so

2^{30} + 2^{23} + 2^{14} = (2^{15} + 2^7)^2

.

The middle term

2^{23}

satisfies

2 \cdot \sqrt{2^{30}} \cdot \sqrt{2^{14}}

.

3

Evaluate the square root and solve for

k

.

\sqrt{(2^{15} + 2^7)^2} = 2^{15} + 128

. Setting

2^{15} + 128 = 2^k + 128

yields

k = 15

.

Comparing terms directly after evaluating

2^7 = 128

isolates

2^k = 2^{15}

.

Key Concept

Application of exponent rules combined with perfect square algebraic identities under radicals

Question 68Question →

How many distinct 4-letter arrangements can be formed by rearranging all of the letters in the word $\text{SEES}$ ?

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Answer: 6

Answer

6

To calculate the number of unique arrangements for a multiset of letters, compute the total factorial divided by the product of factorials for each repeated letter's frequency. For

\text{SEES}

, there are 4 total letters with 2 'S's and 2 'E's, resulting in

\frac{4!}{2!2!} = \frac{24}{4} = 6

.

Step-by-Step Solution

1

Count the total number of letters and identify frequencies of repeated letters.

The word

\text{SEES}

contains 4 letters in total: two 'S's and two 'E's.

To apply the distinct permutations formula, we need the total count of elements and the counts for each repeated identical element.

2

Calculate the number of distinct arrangements using the formula

\frac{n!}{n_1! n_2! \dots n_k!}

.

\frac{4!}{2! \cdot 2!} = \frac{24}{2 \cdot 2} = 6.

Dividing by

2! \cdot 2!

eliminates duplicate counts arising from swapping indistinguishable identical letters.

Key Concept

Permutations with Repetition

Question 69Question →

What is the numerical value of the expression $\sqrt[3]{7 + 5\sqrt{2}} - \sqrt[3]{5\sqrt{2} - 7}$ ?

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Answer: 2

Answer

The numerical value of the expression is 2.

The value of the expression is 2. This can be demonstrated either by setting the expression equal to

x

, cubing both sides to construct the cubic equation

x^3 + 3x - 14 = 0

, and factoring out the real root

x = 2

, or by recognizing that

(1 + \sqrt{2})^3 = 7 + 5\sqrt{2}

and

(\sqrt{2} - 1)^3 = 5\sqrt{2} - 7

, which simplifies the expression directly to

(1 + \sqrt{2}) - (\sqrt{2} - 1) = 2

.

Step-by-Step Solution

1

Define variables for the two cubic terms and write the target expression as a difference.

Let

u = \sqrt[3]{7 + 5\sqrt{2}}

and

v = \sqrt[3]{5\sqrt{2} - 7}

, so the target value is

x = u - v

.

Grouping nested radical terms simplifies the algebraic manipulation.

2

Cube both sides of

x = u - v

using the algebraic identity

(u - v)^3 = u^3 - v^3 - 3uv(u - v)

.

x^3 = u^3 - v^3 - 3uv \cdot x

.

Cubing eliminates the outer radical signs on the cubed terms.

3

Evaluate

u^3 - v^3

and the product

uv

.

u^3 - v^3 = (7 + 5\sqrt{2}) - (5\sqrt{2} - 7) = 14

, and

uv = \sqrt[3]{(5\sqrt{2}+7)(5\sqrt{2}-7)} = \sqrt[3]{50 - 49} = 1

.

Using the difference of squares under the cube root simplifies the product term to 1.

4

Substitute the evaluated terms into the cubic equation and solve for the real root

x

.

x^3 = 14 - 3x \implies x^3 + 3x - 14 = 0 \implies (x - 2)(x^2 + 2x + 7) = 0 \implies x = 2

.

The quadratic factor

x^2 + 2x + 7

has negative discriminant (

4 - 28 = -24

), leaving

x = 2

as the unique real solution.

Key Concept

Simplifying nested radicals using cubic algebraic identities and binomial expansions

Question 70Question →

If $x$ is a real number such that $x^2 - 5x + 1 = 0$ , what is the value of $x^2 + \frac{1}{x^2}$ ?

Show answer & explanation

Answer: 23

Answer

The value of

x^2 + \frac{1}{x^2}

is 23.

Dividing the quadratic equation

x^2 - 5x + 1 = 0

by

x

yields

x + \frac{1}{x} = 5

. Squaring both sides of this identity gives

\left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} = 25

. Subtracting 2 from both sides results in

x^2 + \frac{1}{x^2} = 23

.

Step-by-Step Solution

1

Divide the given quadratic equation

x^2 - 5x + 1 = 0

by

x

x - 5 + \frac{1}{x} = 0 \implies x + \frac{1}{x} = 5

Since

x \neq 0

, dividing by

x

isolates the sum of

x

and its reciprocal.

2

Square both sides of the expression

x + \frac{1}{x} = 5

\left(x + \frac{1}{x}\right)^2 = 25 \implies x^2 + 2 + \frac{1}{x^2} = 25

Expanding the binomial square produces the required quadratic sum along with a constant cross-term.

3

Isolate

x^2 + \frac{1}{x^2}

by subtracting 2 from 25

x^2 + \frac{1}{x^2} = 23

Subtracting the constant cross-term yields the exact requested numeric value.

Key Concept

Algebraic transformation of quadratic equations into reciprocal power sums

Question 71Question →

For any positive integer $n$ , let $S_n$ denote the units digit of the sum $2^n + 3^n + 4^n + 7^n$ . What is the remainder when the sum $T = \sum_{n=1}^{102} S_n$ is divided by $9$ ?

Show answer & explanation

Answer: 1

Answer

The remainder when the sum T is divided by 9 is 1.

Evaluating the units digits of each exponent term reveals that

S_n

follows a repeating 4-term cycle

(6, 8, 2, 4)

with a sum of 20 per cycle. For 102 terms, there are 25 full cycles plus the first two terms (

S_1=6

and

S_2=8

). The total sum is

T = 25 \times 20 + 6 + 8 = 514

. Dividing 514 by 9 yields

514 = 9 \times 57 + 1

, so the remainder is 1.

Step-by-Step Solution

1

Find the cyclicity of the units digit of each term

2^n, 3^n, 4^n,

and

7^n

.

Units digits repeat in patterns of length 4: for

2^n

(2, 4, 8, 6), for

3^n

(3, 9, 7, 1), for

4^n

(4, 6, 4, 6), and for

7^n

(7, 9, 3, 1).

Units digits of positive integer powers cycle with periodicities that divide 4.

2

Compute

S_n

for

n = 1, 2, 3, 4

and find the sum of one 4-term period.

S_1 = \text{units}(2+3+4+7=16) = 6

,

S_2 = \text{units}(4+9+6+9=28) = 8

,

S_3 = \text{units}(8+7+4+3=22) = 2

,

S_4 = \text{units}(6+1+6+1=14) = 4

. Sum of one period =

6+8+2+4 = 20

.

The sum of the units digits of individual terms determines the units digit of the total expression.

3

Calculate the total sum

T = \sum_{n=1}^{102} S_n

.

Since

102 = 25 \times 4 + 2

, the sequence consists of 25 complete cycles of 4 terms plus the first 2 terms (

S_1=6

and

S_2=8

). Thus,

T = 25 \times 20 + 6 + 8 = 500 + 14 = 514

.

Dividing the total number of terms by the period length gives the number of full cycles and remaining initial terms.

4

Compute the remainder when

T = 514

is divided by 9.

The sum of the digits of 514 is

5 + 1 + 4 = 10

, and

10 \equiv 1 \pmod 9

. Alternatively,

514 = 9 \times 57 + 1

. Thus, the remainder is 1.

A positive integer and the sum of its digits leave the same remainder when divided by 9.

Key Concept

Units digit cyclicity of exponential terms and modular arithmetic on sequence sums

Estimated Time:2m 0s

Question 72Question →

Set $S$ consists of $n$ consecutive integers. The sum of the smallest three integers in set $S$ is $-33$ , and the sum of the largest three integers in set $S$ is $69$ . What is the value of $n$ ?

Show answer & explanation

Answer: 37

Answer

The total number of consecutive integers in set

S

is 37.

Solving

3x + 3 = -33

yields a first term of

-12

, and solving

3y - 3 = 69

yields a last term of

24

. The total count of consecutive integers in an inclusive range is

\text{last} - \text{first} + 1

, giving

24 - (-12) + 1 = 37

.

Step-by-Step Solution

1

Find the smallest integer in set S

The smallest integer is -12

Let the smallest integer be

x

. The sum of the smallest three consecutive integers is

x + (x + 1) + (x + 2) = 3x + 3

. Setting

3x + 3 = -33

yields

3x = -36

, so

x = -12

.

2

Find the largest integer in set S

The largest integer is 24

Let the largest integer be

y

. The sum of the largest three consecutive integers is

(y - 2) + (y - 1) + y = 3y - 3

. Setting

3y - 3 = 69

yields

3y = 72

, so

y = 24

.

3

Calculate the total number of elements n in set S

n = 37

For an inclusive set of consecutive integers bounded by first term

a

and last term

b

, the total number of terms is

n = b - a + 1

. Here,

n = 24 - (-12) + 1 = 37

.

Key Concept

Counting inclusive terms in a sequence of consecutive integers

Question 73Question →

A set $S$ consists of $n$ consecutive integers. The sum of all the integers in set $S$ is $675$ , and the product of the smallest integer and the largest integer in set $S$ is $1{}976$ . What is the value of $n$ ?

Show answer & explanation

Answer: 15

Answer

15

For any set of

n

consecutive integers, the median

m

equals the arithmetic mean

\frac{675}{n}

. The smallest and largest elements can be written as

m - \frac{n-1}{2}

and

m + \frac{n-1}{2}

, respectively. Their product is

m^2 - \left(\frac{n-1}{2}\right)^2 = 1976

. Substituting

n = 15

gives

m = 45

, leading to

45^2 - 7^2 = 2025 - 49 = 1976

, which satisfies all conditions.

Step-by-Step Solution

1

Relate the sum of the set to its mean and number of terms

The mean (arithmetic average) of

n

consecutive integers is equal to the median

m

, so

\text{Sum} = n \cdot m = 675

, which implies

m = \frac{675}{n}

.

For any evenly spaced set, the sum equals the number of terms times the mean.

2

Express the smallest and largest elements in terms of the median

m

and number of terms

n

The smallest element is

a = m - \frac{n-1}{2}

and the largest element is

b = m + \frac{n-1}{2}

.

In a set of

n

consecutive integers, the distance from the median to either endpoint is

\frac{n-1}{2}

.

3

Formulate the product equation using the difference of squares

The product of the smallest and largest elements is

a \cdot b = \left(m - \frac{n-1}{2}\right)\left(m + \frac{n-1}{2}\right) = m^2 - \left(\frac{n-1}{2}\right)^2 = 1976

.

Applying the difference of squares identity

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

simplifies the expression.

4

Substitute

m = \frac{675}{n}

and solve for

n

Substituting

m

yields

\left(\frac{675}{n}\right)^2 - \left(\frac{n-1}{2}\right)^2 = 1976

. Testing odd integer factors

n

of

675

:
- If

n = 9

:

m = 75

, product

= 75^2 - 4^2 = 5625 - 16 = 5609 \neq 1976

.
- If

n = 15

:

m = 45

, product

= 45^2 - 7^2 = 2025 - 49 = 1976

.
Hence,

n = 15

.

Since

n

must be a positive integer factor of

675

, checking candidate factors narrows down the unique solution.

Key Concept

Properties of consecutive integer sets: mean-median equivalence and difference of squares decomposition for endpoints.

Alternative Method

Let the set be

\{a, a+1, \dots, a+n-1\}

. The sum is

\frac{n(2a + n - 1)}{2} = 675 \implies n(2a + n - 1) = 1350

. The product of endpoints is

a(a + n - 1) = 1976

. Solving the system of equations for integer values of

a

and

n

yields

a = 38

and

n = 15

.

Estimated Time:2m 30s

Question 74Question →

What is the smallest positive integer $n$ such that $n$ is a multiple of $45$ , $n$ is not divisible by $8$ , and $n$ has exactly $18$ positive integer divisors?

Show answer & explanation

Answer: 180

Answer

180

The smallest positive integer satisfying all conditions is 180 because 180 = 2^2 * 3^2 * 5^1, which has (2+1)(2+1)(1+1) = 18 positive divisors, is a multiple of 45, and is not divisible by 8.

Step-by-Step Solution

1

Determine the prime factorization constraints for n

n must be of the form 2^a * 3^b * 5^c * ..., where b >= 2, c >= 1, and a <= 2.

n is a multiple of 45 = 3^2 * 5^1, requiring at least 3^2 and 5^1. Since n is not divisible by 8 = 2^3, the exponent of 2 cannot exceed 2.

2

Analyze the total number of divisors constraint

(a+1)(b+1)(c+1)... = 18

The total number of positive divisors is calculated by adding 1 to each exponent in the prime factorization and multiplying them together.

3

Find the optimal prime exponent configuration to minimize n

Factor 18 as 3 * 3 * 2, corresponding to exponents 2, 2, and 1.

Using three prime factors (2, 3, and 5) with smaller exponents yields a smaller total integer than using fewer prime factors with larger exponents.

4

Assign exponents to prime bases to yield the minimum integer value

n = 2^2 * 3^2 * 5^1 = 180

Assigning exponent 2 to base 2, exponent 2 to base 3, and exponent 1 to base 5 satisfies all constraints (b = 2 >= 2, c = 1 >= 1, a = 2 < 3) and minimizes the result.

Key Concept

Divisor count formula and prime factorization constraints

Question 75Question →

A bakery sells blueberry muffins for $3 each and chocolate chip muffins for$ 4 each. On Saturday, the bakery sold a total of 50 muffins for $170. How many chocolate chip muffins were sold on Saturday?

Show answer & explanation

Answer: 20

Answer

The bakery sold 20 chocolate chip muffins on Saturday.

Let

b

represent the number of blueberry muffins and

c

represent the number of chocolate chip muffins sold. Based on the problem text, we construct the system of equations:

b + c = 50

and

3b + 4c = 170

. From the first equation,

b = 50 - c

. Substituting this into the second equation yields

3(50 - c) + 4c = 170

, which expands to

150 - 3c + 4c = 170

. Simplifying gives

150 + c = 170

, so

c = 20

.

Step-by-Step Solution

1

Define variables and set up the system of linear equations.

b + c = 50

and

3b + 4c = 170

The total number of muffins sold is 50, and the total revenue from selling blueberry muffins at

3 each and chocolate chip muffins at

4 each is $170.

2

Substitute

b = 50 - c

into the cost equation.

3(50 - c) + 4c = 170 \implies 150 + c = 170

Substituting

b

in terms of

c

creates a single-variable linear equation for

c

.

3

Solve for

c

.

c = 20

Subtracting 150 from both sides gives the exact number of chocolate chip muffins sold.

Key Concept

Solving Systems of Linear Equations by Substitution or Elimination

Question 76Question →

A courier service calculates its total delivery charge using a fixed base fee plus a constant per-mile rate. On Monday, a delivery of 15 miles received a 20% discount on the fixed base fee and a 25% surcharge on the per-mile rate, resulting in a total charge of $30.50. On Tuesday, a delivery of 20 miles incurred a 40% increase on the fixed base fee and received a 15% discount on the per-mile rate, resulting in a total charge of$ 34.40. What is the standard total delivery charge, in dollars, for a 25-mile delivery with no fee adjustments or rate changes?

Show answer & explanation

Answer: 40

Answer

The standard total delivery charge for a 25-mile delivery is 40 dollars.

Translating the scenario into linear equations gives

0.80B + 18.75r = 30.50

and

1.40B + 17.00r = 34.40

. Solving this linear system yields a standard base fee of

B = 10.00

dollars and a standard per-mile rate of

r = 1.20

dollars. Substituting these into the standard 25-mile cost expression

B + 25r

yields

10.00 + 25(1.20) = 40.00

dollars.

Step-by-Step Solution

1

Formulate linear equations from the word problem context.

System of equations:

0.80B + 18.75r = 30.50

and

1.40B + 17.00r = 34.40

.

Applying the percentage adjustments to the fixed base fee

B

and the rate per mile

r

for the given distances yields exact linear expressions.

2

Eliminate variable

B

to solve for

r

.

Multiply equations to equate coefficients of

B

:

5.60B + 131.25r = 213.50

and

5.60B + 68.00r = 137.60

. Subtracting gives

63.25r = 75.90

, so

r = 1.20

.

Finding the per-mile rate

r

allows determination of the standard mileage component.

3

Solve for base fee

B

using

r = 1.20

.

1.40B + 17(1.20) = 34.40 \implies 1.40B = 14.00 \implies B = 10.00

.

Substituting

r

into either linear equation gives the fixed base fee.

4

Calculate the target standard cost for 25 miles.

B + 25r = 10.00 + 25(1.20) = 40.00

.

Evaluating the standard pricing expression

B + 25r

with

B = 10

and

r = 1.20

gives the total cost.

Key Concept

Solving Systems of Two-Variable Linear Equations from Word Problems

Question 77Question →

What is the value of the positive integer $n$ if $n$ is a multiple of 18 and $n$ has exactly 9 positive integer divisors?

Show answer & explanation

Answer: 36

Answer

36

The prime factorization of 18 is

2^1 \cdot 3^2

. Any multiple

n

of 18 must take the form

n = 2^a \cdot 3^b \dots

where

a \ge 1

and

b \ge 2

. The total number of positive divisors of

n

is given by

(a+1)(b+1)\dots = 9

. Given that

a+1 \ge 2

and

b+1 \ge 3

, the only product of integers equal to 9 is

3 \times 3

. This requires

a+1 = 3 \implies a = 2

and

b+1 = 3 \implies b = 2

, with no additional prime factors present. Therefore,

n = 2^2 \cdot 3^2 = 36

.

Step-by-Step Solution

1

Determine the prime factorization constraints imposed by 18.

18 = 2^1 \cdot 3^2

, so

n = 2^a \cdot 3^b \dots

with

a \ge 1

and

b \ge 2

.

Any multiple of 18 must contain at least one factor of 2 and two factors of 3.

2

Apply the divisor count formula to set up an equation.

(a+1)(b+1) = 9

The number of positive divisors of

2^a \cdot 3^b

is given by

(a+1)(b+1)

.

3

Solve for the exponents

a

and

b

.

a = 2

and

b = 2

Because 9 can only be factored as

3 \times 3

for integer components where

a+1 \ge 2

and

b+1 \ge 3

, both

a+1

and

b+1

must equal 3.

4

Compute the value of

n

.

n = 2^2 \cdot 3^2 = 36

Multiply the prime power factors together to find the value of

n

.

Key Concept

The total number of positive divisors of a positive integer

n = p_1^{e_1} p_2^{e_2} \dots p_k^{e_k}

is given by

(e_1 + 1)(e_2 + 1) \dots (e_k + 1)

.

Question 78Question →

If $m$ and $n$ are positive integers such that $2^m - 2^n = 1920$ , what is the value of $m + n$ ?

Show answer & explanation

Answer: 18

Answer

18

To solve

2^m - 2^n = 1920

, factor out

2^n

to express the left side as

2^n(2^{m-n} - 1)

. Prime factorizing 1920 gives

2^7 \times 15

. Since

m

and

n

are positive integers with

m > n

, the term

(2^{m-n} - 1)

is an odd integer. Therefore, the power-of-2 term

2^n

must equal

2^7

, which implies

n = 7

. The odd term

(2^{m-n} - 1)

must equal

15

, leading to

2^{m-n} = 16 = 2^4

, so

m - n = 4

. Solving for

m

gives

m = 11

. Finally,

m + n = 11 + 7 = 18

.

Step-by-Step Solution

1

Factor the exponential expression

2^n(2^{m-n} - 1) = 1920

Factoring out the smaller power of 2 separates the expression into a power of 2 and an odd integer multiplier.

2

Determine the prime factorization of 1920

1920 = 2^7 \times 15

Prime factorization isolates the highest power of 2 (

2^7

) from the remaining odd factor (

15

).

3

Equate corresponding power-of-2 and odd factors

n = 7

and

m - n = 4

The even component

2^n

must equal

2^7

, giving

n = 7

. The odd component

2^{m-n} - 1

must equal

15

, so

2^{m-n} = 16 = 2^4

, giving

m - n = 4

.

4

Solve for

m

and compute

m + n

m = 11

and

m + n = 18

Adding

n = 7

to

m - n = 4

yields

m = 11

. The requested sum is

m + n = 11 + 7 = 18

.

Key Concept

Factoring difference of powers using fundamental exponent rules and equating even/odd prime components.

Question 79Question →

What is the smallest positive integer $n$ such that $n!$ is divisible by $10^{10}$ ?

Show answer & explanation

Answer: 45

Answer

The smallest positive integer

n

such that

n!

is divisible by

10^{10}

is 45.

To find the smallest integer

n

such that

n!

is divisible by

10^{10}

, we need

n!

to contain at least 10 prime factors of 5 (since 2s are abundant). Using Legendre's formula,

E_5(40!) = \lfloor 40/5 \rfloor + \lfloor 40/25 \rfloor = 8 + 1 = 9

, which is insufficient. For

n = 45

,

E_5(45!) = \lfloor 45/5 \rfloor + \lfloor 45/25 \rfloor = 9 + 1 = 10

, satisfying the requirement. Thus, 45 is the smallest positive integer.

Step-by-Step Solution

1

Determine the prime factorization requirement for divisibility by

10^{10}

.

10^{10} = 2^{10} \times 5^{10}

. Thus,

n!

must contain at least 10 factors of 5.

The power of 2 in any factorial

n!

(where

n \ge 5

) is always strictly greater than the power of 5, making 5 the limiting prime factor.

2

Apply Legendre's formula for the exponent of prime

p = 5

in

n!

.

E_5(n!) = \lfloor \frac{n}{5} \rfloor + \lfloor \frac{n}{25} \rfloor + \dots \ge 10

.

Legendre's formula accounts for single multiples of 5, double multiples of 5 (25), etc.

3

Evaluate candidate values for

n

.

For

n = 40

,

E_5(40!) = 8 + 1 = 9

factors. For

n = 45

,

E_5(45!) = 9 + 1 = 10

factors.

Testing multiples of 5 systematically pinpoints the exact boundary where the total count of prime factor 5 reaches 10.

Key Concept

Finding the exponent of a prime factor in a factorial using Legendre's formula

Estimated Time:2m 0s

Question 80Question →

A set $S$ consists of $n$ consecutive positive integers, where $n > 1$ is an odd integer. If the sum of all elements in set $S$ is equal to $3^{10}$ , what is the minimum possible value of the median of set $S$ ?

Show answer & explanation

Answer: 243

Answer

The minimum possible value of the median of set

S

is 243.

For an odd number

n

of consecutive integers, the sum of the set equals

n \times m

, where

m

is the median. Given that

n \times m = 3^{10}

, both

n

and

m

must be powers of 3, so

n = 3^k

and

m = 3^{10-k}

. To ensure all terms in the set are positive, the smallest term

m - \frac{n-1}{2}

must be at least 1, which requires

2m > n

. Substituting the powers of 3 yields

2 \cdot 3^{10-k} > 3^k \implies 3^{2k} < 2 \cdot 3^{10} = 118,098

. The largest integer

k

satisfying this condition is

k = 5

(since

3^{10} = 59,049 < 118,098

while

3^{12} = 531,441 > 118,098

). Maximizing

k

minimizes the median

m = 3^{10-5} = 3^5 = 243

.

Step-by-Step Solution

1

Relate the sum of an evenly spaced set to its number of terms and median.

n \times m = 3^{10}

, where

n

is the number of terms and

m

is the median.

For any set of

n

consecutive integers where

n

is odd, the sum of the set equals the number of terms times the middle term (median).

2

Express

n

and

m

as powers of 3.

n = 3^k

and

m = 3^{10-k}

where

k \ge 1

.

Because

3^{10}

has only 3 as a prime factor, any integer factors

n

and

m

must be powers of 3.

3

Establish the positivity constraint for the terms in set

S

.

2m > n

The smallest term in the set is

m - \frac{n-1}{2}

. Requiring

m - \frac{n-1}{2} \ge 1

gives

2m \ge n + 1

, or strictly

2m > n

.

4

Solve the inequality

2m > n

in terms of

k

.

3^{2k} < 2 \cdot 3^{10} = 118,098

Substituting

n = 3^k

and

m = 3^{10-k}

yields

2 \cdot 3^{10-k} > 3^k

, which rearranges to

3^{2k} < 2 \cdot 3^{10}

.

5

Find the maximum valid integer value of

k

and calculate the corresponding minimum median

m

.

Maximum

k = 5

, giving minimum median

m = 3^{5} = 243

.

For

k = 5

,

3^{10} = 59,049 < 118,098

. For

k = 6

,

3^{12} = 531,441 > 118,098

. Thus

k=5

is the maximum integer

k

, which yields the minimum median

m = 3^{10-5} = 243

.

Key Concept

Sum of consecutive integers set formula and positivity constraints

Answer

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