All practice questions

Question 101Question →

The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two positive three-digit integers $P$ and $Q$ (where $P > Q$ ) are $24$ and $1080$ , respectively. If the difference between the two numbers is $96$ , what is the sum of the two numbers ( $P + Q$ )?

Show answer & explanation

Answer: 336

Answer

The sum of the two numbers P and Q is 336.

By representing

P = 24a

and

Q = 24b

with

\gcd(a, b) = 1

, the relation

\text{HCF} \times \text{LCM} = P \times Q

yields

a \times b = 45

. The coprime factor pairs of

45

are

(45, 1)

and

(9, 5)

. The pair

(45, 1)

gives

1080

and

24

, which are not both three-digit numbers. The pair

(9, 5)

gives

P = 216

and

Q = 120

, both of which are three-digit numbers with a difference of

96

. The sum of these two numbers is

216 + 120 = 336

.

Step-by-Step Solution

1

Express the two numbers in terms of their HCF and coprime factors

Let

P = 24a

and

Q = 24b

where

\gcd(a, b) = 1

and

a > b

.

Any two numbers sharing an HCF of

h

can be represented as

h \cdot a

and

h \cdot b

where

a

and

b

have no common prime factors.

2

Relate the product of the coprime factors to the LCM and HCF

a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{1080}{24} = 45

.

The product of two numbers equals the product of their HCF and LCM:

(24a)(24b) = 24 \times 1080

.

3

Identify all coprime factor pairs of 45

The coprime factor pairs

(a, b)

with

a > b

are

(45, 1)

and

(9, 5)

.

Factor pairs such as

(15, 3)

are invalid because

\gcd(15, 3) = 3 \neq 1

.

4

Apply the three-digit integer and difference constraints to select the valid pair

For

(a, b) = (9, 5)

,

P = 24 \times 9 = 216

and

Q = 24 \times 5 = 120

. Difference =

216 - 120 = 96

.

The pair

(45, 1)

yields

P = 1080

(four digits) and

Q = 24

(two digits), failing the three-digit criteria.

5

Calculate the sum of

P

and

Q

P + Q = 216 + 120 = 336

.

The question asks specifically for the sum

P + Q

.

Key Concept

Properties of HCF and LCM including HCF x LCM = Product of Numbers and Prime Factor Coprimality

Question 102Question →

What is the unit digit of the expression $P = (678^{120} + 359^{45}) \times 243^{37} - 714^{82}$ ?

Show answer & explanation

Answer: 9

Answer

The unit digit of the expression is 9.

Evaluating each term using cyclicity rules gives

678^{120} \rightarrow 6

,

359^{45} \rightarrow 9

,

243^{37} \rightarrow 3

, and

714^{82} \rightarrow 6

. Combining these gives

(6 + 9) \times 3 - 6 \equiv 15 \times 3 - 6 \equiv 5 \times 3 - 6 \equiv 15 - 6 \equiv 9 \pmod{10}

.

Step-by-Step Solution

1

Calculate the unit digit of

678^{120}

Unit digit is 6

The cyclicity of 8 is 4 (8, 4, 2, 6). Exponent 120 mod 4 = 0, which corresponds to the 4th term in the cycle.

2

Calculate the unit digit of

359^{45}

Unit digit is 9

The cyclicity of 9 is 2 (9 for odd powers, 1 for even powers). Exponent 45 is odd.

3

Find the unit digit of

(678^{120} + 359^{45})

Unit digit is 5

Sum of unit digits is 6 + 9 = 15, which has unit digit 5.

4

Calculate the unit digit of

243^{37}

Unit digit is 3

The cyclicity of 3 is 4 (3, 9, 7, 1). Exponent 37 mod 4 = 1, so the unit digit is 3^1 = 3.

5

Multiply the sum by

243^{37}

Unit digit is 5

Product of unit digits is 5 * 3 = 15, which has unit digit 5.

6

Calculate the unit digit of

714^{82}

Unit digit is 6

The cyclicity of 4 is 2 (4 for odd powers, 6 for even powers). Exponent 82 is even.

7

Compute the final unit digit of the overall expression

Unit digit is 9

Subtracting unit digits yields 5 - 6. Adding 10 for borrowing gives 15 - 6 = 9.

Key Concept

Unit Digit and Cyclicity Rules for Exponential Expressions

Estimated Time:1m 30s

Question 103Question →

What is the simplified numerical value of the expression $15 + \left\{ 20 - \left( 8 \div \overline{6 - 2} \right) \right\}$ when evaluated using the standard BODMAS rule?

Show answer & explanation

Answer: 33

Answer

The simplified numerical value of the expression is 33.

Applying VBODMAS order (Vinculum

\rightarrow

Brackets

\rightarrow

Division

\rightarrow

Subtraction

\rightarrow

Addition) yields:

\overline{6-2} = 4

,

8 \div 4 = 2

,

20 - 2 = 18

, and

15 + 18 = 33

.

Step-by-Step Solution

1

Evaluate the expression under the vinculum (bar)

\overline{6 - 2} = 4

The vinculum (bar line) takes precedence over standard operators.

2

Perform division inside the round brackets

8 \div 4 = 2

Operations inside the innermost round brackets are evaluated next.

3

Perform subtraction inside the curly brackets

20 - 2 = 18

Evaluate the expression enclosed in the curly brackets.

4

Perform final addition outside brackets

15 + 18 = 33

Complete the evaluation by adding the remaining outer terms.

Key Concept

BODMAS / VBODMAS Rule (Order of Operations with Vinculum and Brackets)

Question 104Question →

The table below details the renewable energy generation (in Gigawatt-hours, GWh) across four zones in 2025:

Zone	Solar	Wind	Hydro	Biomass
North	120	150	200	50
South	180	220	160	40
East	90	60	250	80
West	210	190	100	70

What is the total solar energy generation (in GWh) across all four zones combined in 2025?

Show answer & explanation

Answer: 600

Answer

The total solar energy generation across all four zones in 2025 is 600 GWh.

Summing the values in the Solar column (

120 + 180 + 90 + 210

) yields a total of

600

GWh.

Step-by-Step Solution

1

Locate and extract the values under the 'Solar' column for all four zones.

North = 120, South = 180, East = 90, West = 210.

The question asks specifically for total solar energy generation.

2

Sum the extracted values across all rows.

120 + 180 + 90 + 210 = 600

GWh.

Combining values across all four zones yields the overall total for the solar category.

Key Concept

Data Extraction and Summation from Tables

Question 105Question →

The following missing data table presents the annual cargo volume (in Thousand Metric Tonnes, TMT) handled across three terminals (Terminal A, Terminal B, and Terminal C) at a major seaport for three distinct cargo categories: Bulk Cargo, Liquid Cargo, and Containerized Cargo.

Terminal	Bulk Cargo	Liquid Cargo	Containerized Cargo	Total
Terminal A	120	—	150	450
Terminal B	—	210	180	500
Terminal C	160	140	—	420
Total	390	530	450	1370

Based on the table above, what is the volume of Liquid Cargo (in Thousand Metric Tonnes) handled by Terminal A?

Show answer & explanation

Answer: 180

Answer

The volume of Liquid Cargo handled by Terminal A is 180 Thousand Metric Tonnes.

The row total for Terminal A is

450\text{ TMT}

. The known components are Bulk Cargo (

120\text{ TMT}

) and Containerized Cargo (

150\text{ TMT}

). Subtracting these from the total yields

450 - (120 + 150) = 180\text{ TMT}

. Alternatively, using the column total for Liquid Cargo (

530\text{ TMT}

) minus the Liquid Cargo of Terminal B (

210\text{ TMT}

) and Terminal C (

140\text{ TMT}

) gives

530 - 350 = 180\text{ TMT}

.

Step-by-Step Solution

1

Examine the row for Terminal A to find total cargo and known cargo types

Total cargo for Terminal A =

450\text{ TMT}

, Bulk Cargo =

120\text{ TMT}

, Containerized Cargo =

150\text{ TMT}

.

The sum of all cargo categories for a terminal equals the row total.

2

Calculate the sum of known cargo categories for Terminal A

120 + 150 = 270\text{ TMT}

.

Combining known entries simplifies finding the unknown value.

3

Subtract the sum of known categories from the row total

450 - 270 = 180\text{ TMT}

.

Liquid Cargo for Terminal A is the remaining portion of its total cargo volume.

4

Cross-verify using the column total for Liquid Cargo

Total Liquid Cargo =

530\text{ TMT}

. Liquid Cargo for Terminal B =

210\text{ TMT}

and Terminal C =

140\text{ TMT}

. Terminal A Liquid Cargo =

530 - (210 + 140) = 530 - 350 = 180\text{ TMT}

.

Cross-verification confirms numerical consistency.

Key Concept

Missing Data Interpretation via Row and Column Equations

Question 106Question →

The table below presents the operational metrics for Agricultural Processing Units across four districts of a state for FY 2025–26:

District	Total Processing Capacity (in '000 MT)	Operational Units Count	Total Raw Produce Received (in '000 MT)	Processing Efficiency Rate (%)
District A	450	18	360	$85\%$
District B	600	24	480	$90\%$
District C	500	20	420	$80\%$
District D	350	14	320	$75\%$

*Note: Processing Efficiency Rate is defined as the percentage of Total Raw Produce Received that is converted into finished products. The remaining portion of received raw produce is lost as processing waste.*

What is the overall average processing waste (in '000 Metric Tonnes) per operational unit across all four districts combined?

Show answer & explanation

Answer: 3.5

Answer

The overall average processing waste per operational unit across all four districts combined is

3.5

thousand Metric Tonnes.

The correct answer is derived by first finding the waste volume for each district based on raw produce received: District A (

360 \times 0.15 = 54

), District B (

480 \times 0.10 = 48

), District C (

420 \times 0.20 = 84

), and District D (

320 \times 0.25 = 80

). Summing these gives a combined total waste of

266

thousand MT. Dividing this total waste by the total count of operational units (

18 + 24 + 20 + 14 = 76

) yields

\frac{266}{76} = 3.5

thousand MT per unit.

Step-by-Step Solution

1

Calculate the processing waste percentage for each district

District A waste rate =

100\% - 85\% = 15\%

; District B waste rate =

100\% - 90\% = 10\%

; District C waste rate =

100\% - 80\% = 20\%

; District D waste rate =

100\% - 75\% = 25\%

.

Processing waste is the complement of processing efficiency rate.

2

Calculate total processing waste generated in each district in '000 MT

District A waste =

360 \times 0.15 = 54

'000 MT; District B waste =

480 \times 0.10 = 48

'000 MT; District C waste =

420 \times 0.20 = 84

'000 MT; District D waste =

320 \times 0.25 = 80

'000 MT.

Absolute waste is obtained by multiplying Total Raw Produce Received by the waste rate.

3

Calculate total waste and total operational units across all four districts combined

Total Waste =

54 + 48 + 84 + 80 = 266

'000 MT. Total Operational Units =

18 + 24 + 20 + 14 = 76

units.

Aggregating metrics across districts is required to find the combined weighted average.

4

Divide total combined waste by total combined operational units

Average Waste per Unit =

\frac{266}{76} = 3.5

'000 MT per unit.

Overall average metric equals total aggregate waste divided by total aggregate count of units.

Key Concept

Weighted Average and Percentage Calculation from Data Grids

Estimated Time:2m 30s

Question 107Question →

Let $N = 2^a \times 3^b \times 7^c$ be a positive integer, where $a$ , $b$ , and $c$ are positive integers. The number of positive factors of $N$ that are divisible by $6$ is $36$ , the number of positive factors of $N$ that are divisible by $14$ is $32$ , and the number of positive factors of $N$ that are divisible by $21$ is $30$ . What is the total number of positive factors of $N$ ?

Show answer & explanation

Answer: 60

Answer

The total number of positive factors of

N

is 60.

By setting up the system of equations based on exponent constraints for each divisor (

6, 14, 21

), we find

a=4, b=3, c=2

. Substituting these into the total factor formula

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

yields

5 \times 4 \times 3 = 60

.

Step-by-Step Solution

1

Formulate algebraic expressions for the constrained factor counts

ab(c+1) = 36

,

ac(b+1) = 32

, and

bc(a+1) = 30

A factor

2^x \times 3^y \times 7^z

is divisible by

6 = 2^1 \times 3^1

if

x \ge 1

,

y \ge 1

,

z \ge 0

, yielding

a \cdot b \cdot (c+1)

factors. Similarly, divisibility by

14

requires

x \ge 1, z \ge 1

, giving

a \cdot c \cdot (b+1)

, and divisibility by

21

requires

y \ge 1, z \ge 1

, giving

b \cdot c \cdot (a+1)

.

2

Expand equations and substitute

P = abc

ab = 36 - P

,

ac = 32 - P

,

bc = 30 - P

Expanding

ab(c+1) = abc + ab = 36

gives

ab = 36 - P

. Applying the same expansion to the other two equations gives expressions for

ac

and

bc

in terms of

P

.

3

Multiply the three equations together and solve for

P

(ab)(ac)(bc) = (abc)^2 = P^2 \Rightarrow (36-P)(32-P)(30-P) = P^2 \Rightarrow P = 24

Testing positive integer values of

P

that divide into products reveals that for

P = 24

,

(36-24)(32-24)(30-24) = 12 \times 8 \times 6 = 576 = 24^2

.

4

Calculate the values of

ab

,

ac

,

bc

, and solve for

a

,

b

,

c

ab = 12

,

ac = 8

,

bc = 6

, giving

a = 4

,

b = 3

,

c = 2

a^2 = \frac{(ab)(ac)}{bc} = \frac{12 \times 8}{6} = 16 \Rightarrow a = 4

. Similarly,

b^2 = \frac{12 \times 6}{8} = 9 \Rightarrow b = 3

, and

c^2 = \frac{8 \times 6}{12} = 4 \Rightarrow c = 2

.

5

Compute the total number of positive factors of

N

(4+1)(3+1)(2+1) = 5 \times 4 \times 3 = 60

The total number of factors of

N = 2^a \times 3^b \times 7^c

is

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

.

Key Concept

Constrained Factor Counting and Prime Factorization Systems

Question 108Question →

If $3^{x + 2} = 81$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is 2.

Rewriting 81 as

3^4

gives

3^{x+2} = 3^4

. Since the bases are equal, setting the exponents equal gives

x + 2 = 4

, which solves to

x = 2

.

Step-by-Step Solution

1

Express the right-hand side with a base of 3

81 = 3^4

Both sides must have the same base to apply the laws of indices.

2

Equate the exponents

x + 2 = 4

If

a^m = a^n

(where

a > 0

and

a \neq 1

), then

m = n

.

3

Solve for

x

x = 2

Subtracting 2 from both sides isolates

x

.

Key Concept

Equating exponents of equal bases

Question 109Question →

What is the unit digit of the expression $K = (377^{123} + 428^{164}) \times 533^{200} - 619^{86}$ ?

Show answer & explanation

Answer: 8

Answer

The unit digit of the expression is 8.

Using cyclicity principles for each power term,

377^{123}

yields unit digit 3 (remainder 3),

428^{164}

yields unit digit 6 (remainder 0

\rightarrow

4th power),

533^{200}

yields unit digit 1 (remainder 0

\rightarrow

4th power), and

619^{86}

yields unit digit 1 (even power of 9). Combining these gives

(3 + 6) \times 1 - 1 = 8

.

Step-by-Step Solution

1

Find the unit digit of

377^{123}

Unit digit is 3

The unit digit of 377 is 7, which follows a cyclicity pattern of 4 (7, 9, 3, 1). Dividing the exponent 123 by 4 leaves a remainder of 3. Thus, the unit digit is given by

7^3 = 343 \rightarrow 3

.

2

Find the unit digit of

428^{164}

Unit digit is 6

The unit digit of 428 is 8, which has a cyclicity of 4 (8, 4, 2, 6). The exponent 164 is divisible by 4 (remainder 0), which corresponds to the 4th position in the cyclicity cycle (

8^4 = 4096 \rightarrow 6

).

3

Evaluate the sum inside the parentheses

(377^{123} + 428^{164})

Unit digit is 9

Adding the individual unit digits gives

3 + 6 = 9

.

4

Find the unit digit of

533^{200}

Unit digit is 1

The unit digit of 533 is 3, which has a cyclicity of 4 (3, 9, 7, 1). Exponent 200 is divisible by 4 (remainder 0), corresponding to the 4th power in the cycle (

3^4 = 81 \rightarrow 1

).

5

Multiply the bracketed sum by

533^{200}

Unit digit is 9

Multiplying the unit digits yields

9 \times 1 = 9

.

6

Find the unit digit of

619^{86}

Unit digit is 1

The unit digit of 619 is 9, which has a cyclicity of 2 (9 for odd powers, 1 for even powers). Since 86 is even, the unit digit is 1.

7

Subtract

619^{86}

from the product

Unit digit is 8

Subtracting the unit digits gives

9 - 1 = 8

.

Key Concept

Unit Digit and Cyclicity

Question 110Question →

A district officer allocates a development fund among four sectors: Health, Education, Infrastructure, and Agriculture. Health receives a fraction equal to $0.1\overline{6}$ of the total fund. Education receives $\frac{3}{8}$ of the remaining fund after the Health allocation. Infrastructure is directly allocated $0.4\overline{3}$ of the original total fund. The remaining amount is given to Agriculture. If the amount allocated to Infrastructure exceeds the amount allocated to Education by ₹ $87,000$ , what is the total development fund in Rupees?

Show answer & explanation

Answer: 720000

Answer

The total development fund is ₹ 720,000.

Converting the recurring decimals gives Health's share as

\frac{1}{6}

and Infrastructure's share as

\frac{13}{30}

. The remaining fund after Health allocation is

\frac{5}{6}

, making Education's share equal to

\frac{3}{8} \times \frac{5}{6} = \frac{5}{16}

of the total. The difference between Infrastructure's and Education's shares is

\frac{13}{30} - \frac{5}{16} = \frac{29}{240}

. Equating

\frac{29}{240}

of the total fund to ₹ 87,000 yields the total fund as ₹ 720,000.

Step-by-Step Solution

1

Convert the recurring decimals into simplified vulgar fractions.

0.1\overline{6} = \frac{16-1}{90} = \frac{15}{90} = \frac{1}{6}

and

0.4\overline{3} = \frac{43-4}{90} = \frac{39}{90} = \frac{13}{30}

.

Converting mixed recurring decimals to fractions allows exact algebraic calculations.

2

Calculate the fraction of the total fund received by Education.

Remaining fund after Health =

1 - \frac{1}{6} = \frac{5}{6}

. Education's fraction =

\frac{3}{8} \times \frac{5}{6} = \frac{5}{16}

.

Education receives a fraction of the remaining fund, not the original total fund.

3

Find the difference between Infrastructure's fraction and Education's fraction.

\frac{13}{30} - \frac{5}{16} = \frac{104 - 75}{240} = \frac{29}{240}

.

The LCM of denominators 30 and 16 is 240.

4

Set up the linear equation with the given monetary difference to find the total fund

F

.

\frac{29}{240} \times F = 87,000 \implies F = \frac{87,000 \times 240}{29} = 3,000 \times 240 = 720,000

.

Solving for

F

yields the original total fund allocation.

Key Concept

Converting mixed recurring decimals to vulgar fractions and evaluating multi-step compound fractional parts of a total quantity.

Question 111Question →

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

45 - \left[ 15 + \left\{ 24 \div \left( 3 + \frac{1}{2} \text{ of } \left( 16 - \overline{8 - 2} \right) \right) \right\} \right]

Show answer & explanation

Answer: 27

Answer

The simplified numerical value of the given expression is 27.

Evaluating step-by-step using VBODMAS order of precedence (Vinculum -> Brackets -> Of -> Division -> Multiplication -> Addition -> Subtraction) yields the simplified result 27.

Step-by-Step Solution

1

Evaluate the vinculum (bar) operation

\overline{8 - 2} = 6

According to VBODMAS, operations under a vinculum take precedence over standard bracket operations.

2

Simplify the innermost subtraction inside the round brackets

16 - 6 = 10

Subtracted the value obtained from the vinculum within the innermost parentheses.

3

Evaluate the 'of' operator

\frac{1}{2} \text{ of } 10 = 5

The 'of' operator represents multiplication with priority immediately following brackets.

4

Complete the addition in the round brackets

3 + 5 = 8

Evaluated the remaining addition within the round brackets.

5

Perform division inside the curly brackets

24 \div 8 = 3

Divided 24 by the simplified result of the round brackets.

6

Perform addition inside the square brackets

15 + 3 = 18

Added 15 to the result from the curly brackets.

7

Perform final subtraction

45 - 18 = 27

Subtracted the square bracket result from 45 to obtain the final simplified value.

Key Concept

Order of Operations (VBODMAS Rule)

Estimated Time:1m 30s

Question 112Question →

If $x = \sqrt{11 + 4\sqrt{6}} - \sqrt{11 - 4\sqrt{6}}$ and $\left(\frac{1}{16}\right)^{x^2 - 15} = 64^{y + 1}$ , what is the value of $y$ ?

Show answer & explanation

Answer: 1

Answer

The value of

y

is

1

.

Simplifying

\sqrt{11 \pm 4\sqrt{6}}

as

\sqrt{8} \pm \sqrt{3}

yields

x = 2\sqrt{3}

and

x^2 = 12

. Substituting

x^2

gives

(1/16)^{-3} = 16^3 = 2^{12}

. Matching exponents with

64^{y+1} = 2^{6y+6}

yields

12 = 6y + 6

, giving

y = 1

.

Step-by-Step Solution

1

Simplify the nested surds for

x

x = 2\sqrt{3}

Express

11 \pm 4\sqrt{6}

as

11 \pm 2\sqrt{24} = (\sqrt{8} \pm \sqrt{3})^2

, giving

\sqrt{11 + 4\sqrt{6}} - \sqrt{11 - 4\sqrt{6}} = (2\sqrt{2} + \sqrt{3}) - (2\sqrt{2} - \sqrt{3}) = 2\sqrt{3}

.

2

Evaluate

x^2

x^2 = 12

Squaring

x = 2\sqrt{3}

yields

x^2 = 4 \times 3 = 12

.

3

Substitute

x^2 = 12

into the exponent of the given equation

\left(\frac{1}{16}\right)^{-3} = 64^{y + 1}

The exponent

x^2 - 15

evaluates to

12 - 15 = -3

.

4

Convert both sides to a common prime base (base 2)

2^{12} = 2^{6y + 6}

\left(\frac{1}{16}\right)^{-3} = 16^3 = (2^4)^3 = 2^{12}

and

64^{y + 1} = (2^6)^{y + 1} = 2^{6y + 6}

.

5

Equate powers and solve for

y

y = 1

Since the bases are identical,

12 = 6y + 6 \implies 6y = 6 \implies y = 1

.

Key Concept

Nested surd evaluation via perfect square conversion combined with solving exponential equations using index rules.

Question 113Question →

In a state disaster relief operation, a total of $1,200$ food packets were distributed among three relief camps: Camp A, Camp B, and Camp C. Camp A received $400$ packets, while Camp B received $150$ more packets than Camp A. The remaining food packets were distributed to Camp C. How many food packets were distributed to Camp C?

Show answer & explanation

Answer: 250

Answer

The total number of food packets distributed to Camp C is 250.

Camp A receives 400 packets. Camp B receives 150 more than Camp A, which equals 400 + 150 = 550 packets. Combined, Camp A and Camp B receive 400 + 550 = 950 packets. Subtracting 950 from the total allocation of 1,200 packets leaves 250 packets for Camp C.

Step-by-Step Solution

1

Identify the number of packets given to Camp A and calculate the packets given to Camp B.

Camp A = 400 packets; Camp B = 400 + 150 = 550 packets.

The paragraph states Camp B received 150 more packets than Camp A.

2

Calculate the total number of packets distributed to Camp A and Camp B together.

Total for Camp A and Camp B = 400 + 550 = 950 packets.

Adding the individual allocations of Camp A and Camp B.

3

Subtract the combined allocation of Camp A and Camp B from the overall total packets.

Camp C = 1200 - 950 = 250 packets.

The remaining packets out of the total 1,200 were given to Camp C.

Key Concept

Data extraction and step-by-step arithmetic computation from narrative caselets

Question 114Question →

What is the unit digit of the numerical expression $M = 3^{105} \times 7^{82} + 8^{96}$ ?

Show answer & explanation

Answer: 3

Answer

The unit digit of the given expression is 3.

The unit digit of an exponential term is determined by its base's cyclicity cycle and the exponent modulo 4. For

3^{105}

,

105 \pmod 4 = 1 \implies 3^1 = 3

. For

7^{82}

,

82 \pmod 4 = 2 \implies 7^2 = 49 \implies 9

. The product of these terms has a unit digit of

(3 \times 9) \pmod{10} = 7

. For

8^{96}

,

96 \pmod 4 = 0

, which indicates the 4th position in the cyclicity pattern of 8 (8, 4, 2, 6), yielding a unit digit of 6. Finally, adding the unit digits gives

(7 + 6) \pmod{10} = 3

.

Step-by-Step Solution

1

Determine the cyclicity and remainder for

3^{105}

Unit digit is 3

The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent by 4 gives

105 \pmod 4 = 1

, corresponding to

3^1 = 3

.

2

Determine the cyclicity and remainder for

7^{82}

Unit digit is 9

The unit digits of powers of 7 repeat in a cycle of 4 (7, 9, 3, 1). Dividing the exponent by 4 gives

82 \pmod 4 = 2

, corresponding to

7^2 = 49

(unit digit 9).

3

Calculate the unit digit of the product

3^{105} \times 7^{82}

Unit digit is 7

Multiply the unit digits of the two factors:

(3 \times 9) = 27

, which has a unit digit of 7.

4

Determine the cyclicity and remainder for

8^{96}

Unit digit is 6

The unit digits of powers of 8 repeat in a cycle of 4 (8, 4, 2, 6). The exponent 96 is completely divisible by 4 (

96 \pmod 4 = 0

). When remainder is 0, we use the 4th power in the cycle (

8^4 = 4096

), which ends in 6.

5

Sum the unit digits of the two main terms

Unit digit is 3

Adding the unit digits gives

(7 + 6) = 13

, which has a unit digit of 3.

Key Concept

Determining the unit digit of composite exponential expressions using power cyclicity.

Question 115Question →

If $x = \sqrt{10 + 2\sqrt{21}} - \sqrt{10 - 2\sqrt{21}}$ and $9^{x^2 - k} = \left(\frac{1}{243}\right)^{k - 6}$ , find the value of $k$ .

Show answer & explanation

Answer: 2

Answer

The value of

k

is 2.

Simplifying

x

yields

x = (\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

, which gives

x^2 = 12

. Substituting

x^2 = 12

into the exponential equation gives

9^{12 - k} = (1/243)^{k - 6}

. Expressing both sides with prime base 3 gives

3^{2(12 - k)} = 3^{-5(k - 6)}

, which simplifies to

3^{24 - 2k} = 3^{30 - 5k}

. Equating the exponents gives

24 - 2k = 30 - 5k

, which solves to

k = 2

.

Step-by-Step Solution

1

Simplify the nested surds for

x

\sqrt{10 + 2\sqrt{21}} = \sqrt{7} + \sqrt{3}

and

\sqrt{10 - 2\sqrt{21}} = \sqrt{7} - \sqrt{3}

Nested surds of the form

\sqrt{a \pm 2\sqrt{b}}

simplify to

\sqrt{m} \pm \sqrt{n}

where

m + n = a

and

m \cdot n = b

.

2

Evaluate

x

and calculate

x^2

x = 2\sqrt{3}

and

x^2 = 12

Subtracting the two surds gives

(\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

. Squaring

2\sqrt{3}

gives

(2)^2 \times 3 = 12

.

3

Express both sides of the exponential equation with base 3

3^{24 - 2k} = 3^{30 - 5k}

Since

9 = 3^2

and

243 = 3^5

,

9^{12 - k} = (3^2)^{12 - k} = 3^{24 - 2k}

and

(1/243)^{k - 6} = (3^{-5})^{k - 6} = 3^{30 - 5k}

.

4

Equate the powers and solve for

k

k = 2

Since bases are equal (

3 > 0, 3 \neq 1

), setting exponents equal gives

24 - 2k = 30 - 5k \implies 3k = 6 \implies k = 2

.

Key Concept

Simplification of nested square root surds

\sqrt{a \pm 2\sqrt{b}}

combined with equating exponents using prime base factorization.

Question 116Question →

Find the unit digit of the numerical expression $N = (1! + 2! + 3! + \dots + 50!)^{25} + 8^{88}$ .

Show answer & explanation

Answer: 9

Answer

The unit digit of the given expression is 9.

All factorials from

5!

to

50!

end in 0, so the sum

(1! + 2! + 3! + \dots + 50!)

ends in

1 + 2 + 6 + 24 = 33

, which has a unit digit of 3. Raising 3 to the power 25 yields a unit digit of 3 because

25 \pmod 4 = 1

. For

8^{88}

, since 88 is divisible by 4 (

88 \pmod 4 = 0

), we take the 4th term of the cyclicity of 8, which is 6. Summing the unit digits yields

3 + 6 = 9

.

Step-by-Step Solution

1

Determine the unit digit of the base sum

(1! + 2! + 3! + \dots + 50!)

.

Unit digit of base is 3.

Factorials

5!

and above all end in 0 (

5! = 120

), so only

1! + 2! + 3! + 4! = 33

contributes to the unit digit.

2

Calculate the unit digit of

3^{25}

.

Unit digit is 3.

The cyclicity pattern of 3 has period 4 (3, 9, 7, 1). Exponent

25 \pmod 4 = 1

, giving

3^1 = 3

.

3

Calculate the unit digit of

8^{88}

.

Unit digit is 6.

The cyclicity pattern of 8 has period 4 (8, 4, 2, 6). Exponent

88 \pmod 4 = 0

, so we take the 4th position in the cycle, which is 6.

4

Add the unit digits of both terms.

Final unit digit is 9.

3 + 6 = 9

, which gives a unit digit of 9.

Key Concept

Cyclicity of numbers and factorial unit digit properties

Question 117Question →

Under a State Skill Development Mission, a total of $4,500$ candidates were enrolled across three vocational sectors: IT & Electronics, Healthcare & Allied Sciences, and Textile & Apparel. The IT & Electronics sector accounted for $40\%$ of the total enrolled candidates. The ratio of candidates enrolled in Healthcare & Allied Sciences to those in Textile & Apparel was $7 : 8$ . Upon completion of the training, $65\%$ of the enrolled candidates in IT & Electronics successfully obtained certification. The number of certified candidates in Healthcare & Allied Sciences was equal to $80\%$ of the certified candidates in IT & Electronics. If the total number of certified candidates across all three sectors combined was $2,898$ , what percentage of the enrolled candidates in the Textile & Apparel sector successfully obtained certification?

Show answer & explanation

Answer: 55

Answer

The certification percentage of enrolled candidates in the Textile & Apparel sector is

55\%

.

To find the certification percentage for the Textile & Apparel sector, first determine the enrollment figures: IT & Electronics has

1,800

candidates (

40\%

of

4,500

), leaving

2,700

candidates split in a

7 : 8

ratio between Healthcare (

1,260

) and Textile (

1,440

). Next, calculate certified candidates: IT & Electronics has

1,170

certified (

65\%

of

1,800

), and Healthcare has

936

certified (

80\%

of

1,170

). Subtracting these from the total

2,898

certified candidates leaves

792

certified candidates in Textile & Apparel. Finally,

\frac{792}{1,440} \times 100\% = 55\%

.

Step-by-Step Solution

1

Calculate the number of candidates enrolled in IT & Electronics

Enrolled in IT & Electronics =

0.40 \times 4,500 = 1,800

candidates.

IT & Electronics accounts for

40\%

of the overall

4,500

candidates.

2

Determine enrollment in Healthcare & Allied Sciences and Textile & Apparel

Remaining candidates =

4,500 - 1,800 = 2,700

. Enrolled in Healthcare =

\frac{7}{15} \times 2,700 = 1,260

. Enrolled in Textile =

\frac{8}{15} \times 2,700 = 1,440

.

The remaining

2,700

candidates are distributed between Healthcare and Textile in the ratio

7 : 8

(total

15

parts).

3

Calculate certified candidates in IT & Electronics

Certified in IT & Electronics =

0.65 \times 1,800 = 1,170

candidates.

65\%

of the enrolled IT & Electronics candidates achieved certification.

4

Calculate certified candidates in Healthcare & Allied Sciences

Certified in Healthcare =

0.80 \times 1,170 = 936

candidates.

Healthcare certified candidates equal

80\%

of certified candidates in IT & Electronics.

5

Determine certified candidates in Textile & Apparel

Certified in Textile =

2,898 - (1,170 + 936) = 2,898 - 2,106 = 792

candidates.

Total certified across all sectors is

2,898

.

6

Compute the certification rate for Textile & Apparel

Certification percentage =

\left(\frac{792}{1,440}\right) \times 100\% = 55\%

.

Divide certified Textile candidates by total enrolled Textile candidates and convert to percentage.

Key Concept

Data extraction, ratio partitioning, and sequential percentage calculation from unstructured text caselets.

Question 118Question →

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

12.5 \times 4 - \left[ 16 + \left\{ 25 \div \left( 3.5 + \overline{2.8 - 1.3} \right) \right\} \right]

Show answer & explanation

Answer: 29

Answer

The simplified numerical value of the expression is 29.

Applying VBODMAS step-by-step resolves the vinculum first (2.8 - 1.3 = 1.5), then the round brackets (3.5 + 1.5 = 5), curly brackets (25 / 5 = 5), square brackets (16 + 5 = 21), and finally the multiplication and subtraction (50 - 21 = 29).

Step-by-Step Solution

1

Evaluate the vinculum (bar) operation first

\overline{2.8 - 1.3} = 1.5

According to VBODMAS, the vinculum takes top priority over all other operations.

2

Simplify the expression inside the innermost round brackets

3.5 + 1.5 = 5

Next in hierarchy are the round brackets (parentheses).

3

Simplify the expression inside the curly brackets

25 \div 5 = 5

Evaluate operations inside curly braces next.

4

Simplify the expression inside the square brackets

16 + 5 = 21

Complete the evaluation of the outer square brackets.

5

Perform the multiplication and final subtraction outside the brackets

12.5 \times 4 = 50 \text{ and then } 50 - 21 = 29

Perform multiplication before final subtraction following standard operator precedence.

Key Concept

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

Estimated Time:1m 30s

Question 119Question →

Let $N = 2^5 \times 3^4 \times 5^2$ . How many positive integer factors of $N$ are divisible by $12$ but not divisible by $72$ ?

Show answer & explanation

Answer: 21

Answer

21

The correct answer is 21. By analyzing the exponent bounds of

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

, there are 48 factors of

N

divisible by

12

(

a \ge 2, b \ge 1

). Among these, 27 factors are also divisible by

72

(

a \ge 3, b \ge 2

). Subtracting the two counts gives

48 - 27 = 21

factors divisible by 12 but not by 72.

Step-by-Step Solution

1

Represent the general form of a factor of

N

Every positive factor of

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

can be uniquely written as

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

, where

a \in \{0, 1, 2, 3, 4, 5\}

,

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

, and

c \in \{0, 1, 2\}

.

By the Fundamental Theorem of Arithmetic, any factor of a number in prime-factored form takes exponents bounded by the original number's exponents.

2

Identify exponent constraints for divisibility by 12

Since

12 = 2^2 \times 3^1

, for

f

to be divisible by

12

, we must have

a \ge 2

and

b \ge 1

.

Divisibility requires prime factor exponents in the factor to be at least as large as those in the divisor.

3

Calculate the total number of factors divisible by 12

Number of choices for

a

:

5 - 2 + 1 = 4

(values

2, 3, 4, 5

). Number of choices for

b

:

4 - 1 + 1 = 4

(values

1, 2, 3, 4

). Number of choices for

c

:

2 - 0 + 1 = 3

(values

0, 1, 2

). Total count =

4 \times 4 \times 3 = 48

.

By the multiplication principle of counting, independent choices for each exponent are multiplied.

4

Identify exponent constraints and count factors divisible by 72

Since

72 = 2^3 \times 3^2

, a factor is divisible by

72

if

a \ge 3

and

b \ge 2

. Number of choices:

a \in \{3, 4, 5\}

(3 choices),

b \in \{2, 3, 4\}

(3 choices),

c \in \{0, 1, 2\}

(3 choices). Total count =

3 \times 3 \times 3 = 27

.

Every factor of

N

that is divisible by 72 is inherently also a factor divisible by 12, forming a strict subset.

5

Apply the principle of subtraction to find the required count

Required count = (Factors divisible by 12) - (Factors divisible by 72) =

48 - 27 = 21

.

The set of factors divisible by 12 but not by 72 is the set difference between factors divisible by 12 and factors divisible by 72.

Key Concept

Counting Factors with Divisibility Constraints using Prime Factorization

Estimated Time:2m 0s

Question 120Question →

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

Answer

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