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Question 81Question →

An inventory tracking system logs the total number of specialized microchips stored in a warehouse as exactly 5×1433−3×28505 \times 14^{33} - 3 \times 28^{50} units. These microchips are being shipped out in standard crates that hold exactly 1313 units each. The remaining microchips that do not make a full crate will be set aside for quality assurance testing. How many microchips will be set aside?

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Answer: 6

Answer

6 microchips will be set aside for quality assurance testing.
By evaluating the modulo 1313 for each term independently using exponent cyclicity, the expression 5×14335 \times 14^{33} reduces to 55, and 3×28503 \times 28^{50} reduces to 1212. Subtracting these yields an intermediate remainder of −7-7. Since remainders representing physical quantities must be non-negative, we add the divisor (1313) to −7-7, resulting in a final positive remainder of 66.

Step-by-Step Solution

1
Formulate the modulo expression.
(5×1433−3×2850)(mod13)(5 \times 14^{33} - 3 \times 28^{50}) \pmod{13}
Finding the remaining chips after filling crates of 13 is equivalent to calculating the remainder when the total is divided by 13.
2
Reduce the base of the first term.
5×133=5(mod13)5 \times 1^{33} = 5 \pmod{13}
1414 leaves a remainder of 11 when divided by 1313. Any power of 11 remains 11.
3
Reduce the base of the second term.
3×250(mod13)3 \times 2^{50} \pmod{13}
2828 divided by 1313 is 22 with a remainder of 22.
4
Apply Fermat's Little Theorem to evaluate 250(mod13)2^{50} \pmod{13}.
212≡1(mod13)2^{12} \equiv 1 \pmod{13}, so 250≡22≡4(mod13)2^{50} \equiv 2^2 \equiv 4 \pmod{13}
Powers of 22 repeat every 1212 modulo 1313. Since 50=12×4+250 = 12 \times 4 + 2, the remainder of the exponent is 22.
5
Evaluate the full second term.
3×4=12(mod13)3 \times 4 = 12 \pmod{13}
Multiply the evaluated exponent component by the original coefficient.
6
Subtract to find the initial remainder.
5−12=−75 - 12 = -7
Substitute the evaluated remainders back into the full expression.
7
Convert the negative remainder into a positive value.
−7+13=6-7 + 13 = 6
A physical count must be a positive integer. Adding the divisor to a negative remainder gives the true positive remainder.

Key Concept

Modular arithmetic properties, Fermat's Little Theorem, and conversion of negative remainders.
Question 82Question →

A civil engineering team is installing three different types of sensor cables along a newly constructed bridge. The standard supplier rolls for these cables come in lengths of 552\frac{55}{2} meters, 774\frac{77}{4} meters, and 1215\frac{121}{5} meters. To ensure modular replacement without any waste, the team must cut all the rolls into smaller, equal-length segments. What is the maximum possible length, in meters, of each segment?

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Answer: 0.55

Answer

0.55 meters
The problem asks for the maximum equal length that can divide all three roll lengths without any remainder, which mathematically corresponds to the Highest Common Factor (HCF). Using the standard formula for fractions: HCF = HCF(55, 77, 121) / LCM(2, 4, 5) = 11 / 20 = 0.55 meters.

Step-by-Step Solution

1
Determine the correct mathematical operation based on the problem context.
The requirement to cut multiple rolls into equal segments of maximum possible length with zero waste indicates the need to find the Highest Common Factor (HCF) of the three lengths.
HCF is used to find the largest common quantity that can perfectly divide multiple numbers.
2
State the standard formula for finding the HCF of fractions.
HCF=HCF of numeratorsLCM of denominatorsHCF = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}
This is the mathematical rule for finding the HCF of rational numbers.
3
Calculate the HCF of the numerators.
The numerators are 5555, 7777, and 121121. Their prime factorizations are 5×115 \times 11, 7×117 \times 11, and 11×1111 \times 11. The HCF is 1111.
The largest common prime factor across all three numerators is 11.
4
Calculate the LCM of the denominators.
The denominators are 22, 44, and 55. The lowest common multiple is 2020.
20 is the smallest positive integer divisible by 2, 4, and 5.
5
Compute the final decimal value.
1120=0.55\frac{11}{20} = 0.55
Converting the fraction to a decimal yields the exact required length.

Key Concept

Highest Common Factor (HCF) of Fractions
Question 83Question →

A field researcher starts from a central station, initially facing North. He walks 12 m12\text{ m} due North, turns 90∘90^\circ clockwise, and walks 9 m9\text{ m}. He then turns 135∘135^\circ clockwise and walks 152 m15\sqrt{2}\text{ m} straight. From this position, he turns 45∘45^\circ anti-clockwise and walks 5 m5\text{ m}. Finally, he turns 90∘90^\circ anti-clockwise and walks 12 m12\text{ m} to reach a monitoring post. What is the shortest straight-line distance (in meters) between the central station and the monitoring post?

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Answer: 10

Answer

The shortest straight-line distance between the central station and the monitoring post is 10 m10\text{ m}.
By resolving each displacement into xx (East-West) and yy (North-South) components, the final position relative to the starting central station is (6,−8)(6, -8). Applying the Pythagorean theorem 62+(−8)2\sqrt{6^2 + (-8)^2} yields 10 m10\text{ m}.

Step-by-Step Solution

1
Represent the movements as vector displacements on a Cartesian coordinate plane with origin (0,0)(0, 0) at the central station.
Initial position is (0,0)(0, 0) facing North.
Cartesian grid helps systematically evaluate multi-turn directional paths.
2
Calculate position after leg 1 and leg 2.
Leg 1 (12 m North): position is (0,12)(0, 12), facing North. Leg 2 (90° clockwise = East, 9 m): position is (9,12)(9, 12), facing East.
Moving North increases yy-coordinate; turning 90° clockwise points East, increasing xx-coordinate.
3
Calculate position after leg 3.
Turning 135° clockwise from East sets direction to South-West (225∘225^\circ). Displacement vector is (−152cos⁡45∘,−152sin⁡45∘)=(−15,−15)(-15\sqrt{2} \cos 45^\circ, -15\sqrt{2} \sin 45^\circ) = (-15, -15). New position is (9−15,12−15)=(−6,−3)(9 - 15, 12 - 15) = (-6, -3), facing South-West.
135° clockwise turn from East (90°) points along the South-West diagonal.
4
Calculate position after leg 4.
Turning 45° anti-clockwise from South-West sets direction to South (180∘180^\circ). Walking 5 m South gives new position (−6,−3−5)=(−6,−8)(-6, -3 - 5) = (-6, -8), facing South.
45° anti-clockwise turn from South-West (225°) rotates direction to South (180°).
5
Calculate position after leg 5.
Turning 90° anti-clockwise from South sets direction to East (90∘90^\circ). Walking 12 m East gives final position (−6+12,−8)=(6,−8)(-6 + 12, -8) = (6, -8).
90° anti-clockwise turn from South rotates direction to East, increasing xx-coordinate by 12.
6
Compute the shortest straight-line distance from origin (0,0)(0, 0) to final position (6,−8)(6, -8).
Distance d=62+(−8)2=36+64=100=10 md = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ m}.
Shortest distance between two points on a plane is given by the Pythagorean distance formula.

Key Concept

Shortest distance calculation using 2D vector coordinate mapping and angle turns
Estimated Time:1m 30s
Question 84Question →
Determine the next term that logically completes the following numerical sequence:
4,7,14,27,48,79,…4, 7, 14, 27, 48, 79, \dots
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Answer: 122

Answer

The next term in the sequence is 122.
The sequence follows a double-difference pattern. The differences between successive terms are 3, 7, 13, 21, 31. Taking the differences of these values gives 4, 6, 8, 10, which increases by 2 each time. The next second-order difference must be 12, making the next first-order difference 31 + 12 = 43. Adding 43 to 79 gives the final result of 122.

Step-by-Step Solution

1
Calculate the first-order differences between consecutive terms
The first differences are 3, 7, 13, 21, and 31.
Identifying the rate of increase helps determine the underlying rule.
2
Calculate the second-order differences
The second differences are 4, 6, 8, and 10.
Since the first differences are not constant, analyzing second-level differences reveals higher-order patterns.
3
Extend the second-difference pattern to find the next first difference and term
Next second difference = 12, next first difference = 43, next term = 79 + 43 = 122.
The second differences form an arithmetic sequence adding 2 at each step.

Key Concept

Double-Difference Numerical Series
Question 85Question →
Consider the following numerical sequence:
5,14,19,54,83,…5, 14, 19, 54, 83, \dots
What is the value of the next term in this sequence?
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Answer: 202

Answer

The next term in the numerical sequence is 202.
The sequence follows the recurrence pattern an=2an−1+(−1)nn2a_n = 2 a_{n-1} + (-1)^n n^2 starting with a1=5a_1 = 5. To find the 6th term, double the 5th term (83×2=16683 \times 2 = 166) and add 62=366^2 = 36, resulting in 166+36=202166 + 36 = 202.

Step-by-Step Solution

1
Identify the operation connecting the first two terms (55 and 1414).
a2=2×a1+22=2(5)+4=14a_2 = 2 \times a_1 + 2^2 = 2(5) + 4 = 14
Establishes the initial multiplier and square term offset.
2
Determine the rule for the third, fourth, and fifth terms.
a3=2(14)−32=19a_3 = 2(14) - 3^2 = 19, a4=2(19)+42=54a_4 = 2(19) + 4^2 = 54, a5=2(54)−52=83a_5 = 2(54) - 5^2 = 83
Shows an alternating addition and subtraction of consecutive squares n2n^2 after doubling the previous term.
3
Apply the rule for n=6n=6 to calculate the missing term.
a6=2(83)+62=166+36=202a_6 = 2(83) + 6^2 = 166 + 36 = 202
Since the 5th term subtracted 525^2, the 6th term must add 626^2 to twice the 5th term.

Key Concept

Alternating Operational and Square Offsets in Recursive Series
Question 86Question →

A transportation company schedules its daily long-distance routes by increasing the total daily mileage according to a specific mathematical progression. Over the first five days of a new expansion phase, the daily total route distances recorded are 55, 1212, 2626, 4949, and 8383 miles respectively.

If this pattern continues strictly, what will be the total route distance in miles for the sixth day?

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Answer: 130

Answer

130
The sequence follows a double-difference pattern where the second-level differences form a simple arithmetic progression increasing by 22. By extending this pattern, the next addition to the sequence is 4747, which yields exactly 130130 when added to the fifth term.

Step-by-Step Solution

1
Calculate the differences between consecutive terms in the given sequence.
12−5=712 - 5 = 7; 26−12=1426 - 12 = 14; 49−26=2349 - 26 = 23; 83−49=3483 - 49 = 34.
Finding the first-level differences is the standard approach to identify hidden polynomial sequences.
2
Calculate the differences of the first-level differences (second-level differences).
14−7=714 - 7 = 7; 23−14=923 - 14 = 9; 34−23=1134 - 23 = 11.
The first-level differences do not form a constant sequence, but measuring their rate of change reveals an underlying arithmetic progression.
3
Identify the pattern in the second-level differences and extrapolate the next value.
The second-level differences (77, 99, 1111) increase by 22 each time. The next second-level difference will be 11+2=1311 + 2 = 13.
Establishing the core constant pattern allows for accurate forward prediction.
4
Calculate the next first-level difference and the final 6th term.
Next first-level difference = 34+13=4734 + 13 = 47. Sixth term = 83+47=13083 + 47 = 130.
Applying the extrapolated differences forward up the chain yields the final required value.

Key Concept

Double Difference Number Series
Question 87Question →

Consider the following numerical sequence: 11,14,23,44,83,…11, 14, 23, 44, 83, \dots What is the next term in this sequence?

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Answer: 146

Answer

The next term in the sequence is 146.
The sequence follows a double-difference pattern where the differences between consecutive terms (3,9,21,393, 9, 21, 39) themselves differ by an arithmetic progression of multiples of 6 (6,12,186, 12, 18). Adding the next increment of 24 to 39 gives a term difference of 63. Adding 63 to the last term 83 results in 146.

Step-by-Step Solution

1
Find the first difference between consecutive terms
The differences are 3, 9, 21, and 39.
Analyzing differences helps reveal underlying polynomial or double-difference patterns.
2
Find the second difference (difference of differences)
The second differences are 6, 12, and 18.
The first differences do not form an obvious basic sequence, so taking second differences reveals the hidden pattern.
3
Determine the next first-difference value
Next second difference is 24, making the next first difference 39 + 24 = 63.
The second-level differences increase by 6 each step (multiples of 6).
4
Calculate the next term in the sequence
83 + 63 = 146
Adding the computed term difference to the latest term yields the next sequence number.

Key Concept

Double Difference Number Series
Estimated Time:1m 30s
Question 88Question →
Determine the value of the missing term that logically continues the given numerical sequence:
6,13,28,59,122,?6, 13, 28, 59, 122, ?
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Answer: 249

Answer

The missing term in the sequence is 249.
The sequence follows the pattern Tn=Tn−1×2+(n−1)T_n = T_{n-1} \times 2 + (n-1) for n≥2n \ge 2. Thus, the 6th term is 122×2+5=249122 \times 2 + 5 = 249.

Step-by-Step Solution

1
Examine the transition between consecutive numbers in the series
From 6 to 13: 6×2+1=136 \times 2 + 1 = 13
From 13 to 28: 13×2+2=2813 \times 2 + 2 = 28
Establishing the basic recursive function pattern
2
Verify the pattern across remaining terms
From 28 to 59: 28×2+3=5928 \times 2 + 3 = 59
From 59 to 122: 59×2+4=12259 \times 2 + 4 = 122
Confirming that the multiplier is constant (2) while the added constant increments by 1 at each step
3
Apply the rule to find the missing term
122×2+5=244+5=249122 \times 2 + 5 = 244 + 5 = 249
Extending the pattern by adding 5 after doubling the previous term

Key Concept

Recursive series with constant multiplier and arithmetic additive offset
Estimated Time:1m 0s
Question 89Question →
Evaluate the following mathematical expression by applying the standard order of operations (BODMAS):
18÷[412−{214+34×(4.8−2.4+0.4‾)}]+5.518 \div \left[ 4 \frac{1}{2} - \left\{ 2 \frac{1}{4} + \frac{3}{4} \times \left( 4.8 - \overline{2.4 + 0.4} \right) \right\} \right] + 5.5
What is the final numerical value of the simplified expression?
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Answer: 29.5

Answer

The simplified numerical value of the given expression is 29.5.
Following the BODMAS rule systematically, we first resolve the vinculum to get 2.8, then evaluate the round brackets (4.8 - 2.8 = 2.0). Next, inside the curly brackets, multiplication yields 1.5 and addition yields 3.75. Evaluating the square brackets yields 4.5 - 3.75 = 0.75. Dividing 18 by 0.75 gives 24, and adding 5.5 gives the final correct answer of 29.5.

Step-by-Step Solution

1
Evaluate the expression under the vinculum bar
\overline{2.4 + 0.4} = 2.8
The vinculum (bar) takes highest priority over standard parentheses and operations.
2
Simplify the innermost round brackets
4.8 - 2.8 = 2.0
Perform subtraction inside the round brackets next.
3
Perform multiplication inside the curly brackets
34×2.0=1.5\frac{3}{4} \times 2.0 = 1.5
Multiplication precedes addition within the curly brackets according to BODMAS.
4
Complete evaluation inside the curly brackets
2 \frac{1}{4} + 1.5 = 2.25 + 1.5 = 3.75
Convert the mixed fraction 2 1/4 to decimal 2.25 and add 1.5.
5
Evaluate the terms inside the square brackets
4 \frac{1}{2} - 3.75 = 4.5 - 3.75 = 0.75
Convert mixed fraction 4 1/2 to decimal 4.5 and subtract 3.75.
6
Perform division before final addition
18÷0.75=2418 \div 0.75 = 24
Division takes precedence over the final addition operation.
7
Add the final constant
24 + 5.5 = 29.5
Complete the evaluation by adding 5.5 to 24.

Key Concept

Hierarchy of operations in BODMAS including vinculum bar, nested brackets, and precedence of operations.
Estimated Time:1m 30s
Question 90Question →
Calculate the numerical value of the following mathematical expression:
75% of 240−[42÷{214+(3.5−1.8−0.8‾)×1.5}]75\% \text{ of } 240 - \left[ 42 \div \left\{ 2\frac{1}{4} + \left( 3.5 - \overline{1.8 - 0.8} \right) \times 1.5 \right\} \right]
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Answer: 173

Answer

The simplified numerical value of the expression is 173.
Applying the VBODMAS rule sequentially (vinculum, round brackets, multiplication, addition within curly brackets, square bracket division, percentage evaluation, and final subtraction) yields 173.

Step-by-Step Solution

1
Evaluate the vinculum bar expression
1.8−0.8‾=1.0\overline{1.8 - 0.8} = 1.0
Under VBODMAS rules, the vinculum (bar) grouping has the highest priority and must be evaluated first.
2
Evaluate the subtraction inside the round brackets
3.5−1.0=2.53.5 - 1.0 = 2.5
Evaluate the parentheses next using the result from the vinculum evaluation.
3
Perform multiplication inside the curly brackets
2.5×1.5=3.752.5 \times 1.5 = 3.75
Within the curly brackets, multiplication precedes addition according to BODMAS.
4
Perform addition inside the curly brackets
214+3.75=2.25+3.75=6.02\frac{1}{4} + 3.75 = 2.25 + 3.75 = 6.0
Convert the mixed fraction 2142\frac{1}{4} to decimal 2.252.25 and add to 3.753.75.
5
Perform division inside the square brackets
42÷6.0=742 \div 6.0 = 7
Resolve the square brackets by dividing 42 by the value of the curly bracket expression.
6
Evaluate the percentage 'of' expression
75% of 240=0.75×240=18075\% \text{ of } 240 = 0.75 \times 240 = 180
The 'of' operator indicates multiplication applied to percentage.
7
Subtract the square bracket result from the percentage value
180−7=173180 - 7 = 173
Perform final subtraction to complete the calculation.

Key Concept

Hierarchical priority of operations in VBODMAS (Vinculum -> Brackets -> Of -> Division -> Multiplication -> Addition -> Subtraction)
Question 91Question →

Determine the exact result of the given mathematical expression after applying the standard order of operations (BODMAS):

32.4−45×[45÷{2.5×4−(7.6−4.1−1.5‾)}]32.4 - \frac{4}{5} \times \left[ 45 \div \left\{ 2.5 \times 4 - \left( 7.6 - \overline{4.1 - 1.5} \right) \right\} \right]
Show answer & explanation

Answer: 25.2

Answer

The simplified numerical value of the expression is 25.2.
Following the standard BODMAS priority order: evaluate the vinculum (4.1 - 1.5 = 2.6), round brackets (7.6 - 2.6 = 5), curly brackets (2.5 * 4 - 5 = 5), square brackets (45 / 5 = 9), multiplication ((4/5) * 9 = 7.2), and finally subtraction (32.4 - 7.2 = 25.2).

Step-by-Step Solution

1
Evaluate the sub-expression under the vinculum bar
\overline{4.1 - 1.5} = 2.6
The vinculum acts as the innermost grouping symbol and takes highest priority.
2
Simplify the expression inside the round brackets
(7.6 - 2.6) = 5.0
Parentheses (round brackets) are evaluated next.
3
Evaluate operations inside the curly brackets using operator precedence
\{2.5 \times 4 - 5.0\} = \{10 - 5.0\} = 5
Multiplication precedes subtraction inside the curly brackets.
4
Evaluate the division inside the square brackets
[45÷5]=9[45 \div 5] = 9
Square brackets are solved after resolving inner curly brackets.
5
Perform multiplication outside the bracket
\frac{4}{5} \times 9 = 0.8 \times 9 = 7.2
Multiplication takes precedence over final addition/subtraction.
6
Perform the final subtraction
32.4 - 7.2 = 25.2
Subtraction is the final step in the BODMAS order of operations.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets
Estimated Time:1m 30s
Question 92Question →

What is the unit digit of the expression K=(23385×43794)−16847K = (233^{85} \times 437^{94}) - 168^{47}?

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Answer: 5

Answer

The unit digit of the given expression is 5.
To find the unit digit of K=(23385×43794)−16847K = (233^{85} \times 437^{94}) - 168^{47}, analyze each term using base cyclicity rules:
1. 23385233^{85}: Base digit 3 has cyclicity 4. 85÷4=2185 \div 4 = 21 remainder 1. 31=33^1 = 3.
2. 43794437^{94}: Base digit 7 has cyclicity 4. 94÷4=2394 \div 4 = 23 remainder 2. 72=49→97^2 = 49 \rightarrow 9.
3. Product unit digit: 3×9=27→73 \times 9 = 27 \rightarrow 7.
4. 16847168^{47}: Base digit 8 has cyclicity 4. 47÷4=1147 \div 4 = 11 remainder 3. 83=512→28^3 = 512 \rightarrow 2.
5. Subtracting gives 7−2=57 - 2 = 5.

Step-by-Step Solution

1
Find the unit digit of 23385233^{85}
Unit digit is 3
The unit digit of powers of 3 follows a cycle of 4: (3, 9, 7, 1). Since 85≡1(mod4)85 \equiv 1 \pmod 4, the unit digit is 31=33^1 = 3.
2
Find the unit digit of 43794437^{94}
Unit digit is 9
The unit digit of powers of 7 follows a cycle of 4: (7, 9, 3, 1). Since 94≡2(mod4)94 \equiv 2 \pmod 4, the unit digit is 72=49  ⟹  97^2 = 49 \implies 9.
3
Find the unit digit of the product (23385×43794)(233^{85} \times 437^{94})
Unit digit is 7
Multiplying the unit digits of the two terms gives 3×9=273 \times 9 = 27, which has a unit digit of 7.
4
Find the unit digit of 16847168^{47}
Unit digit is 2
The unit digit of powers of 8 follows a cycle of 4: (8, 4, 2, 6). Since 47≡3(mod4)47 \equiv 3 \pmod 4, the unit digit is 83=512  ⟹  28^3 = 512 \implies 2.
5
Calculate the final unit digit of the composite expression
5
Subtracting the unit digit of the subtracted term from the product's unit digit yields 7−2=57 - 2 = 5.

Key Concept

Unit Digit and Cyclicity
Estimated Time:1m 30s
Question 93Question →

Determine the unit digit of the finite series and exponent tower expression S=∑k=150(k!)k!+777S = \sum_{k=1}^{50} (k!)^{k!} + 7^{7^7}.

Show answer & explanation

Answer: 0

Answer

The unit digit of the expression is 0.
The sum of the factorial terms from k=1 to k=4 contributes unit digits of 1, 4, 6, and 6, totaling 17 (unit digit 7). Terms for k >= 5 end in 0. The power tower term 7^(7^7) has an exponent 7^7 which leaves a remainder of 3 when divided by 4. Since the cyclicity of 7 is 4, 7^3 gives a unit digit of 3. Adding 7 and 3 results in 10, giving a final unit digit of 0.

Step-by-Step Solution

1
Calculate unit digits of factorial terms for k from 1 to 4
Unit digits are 1, 4, 6, and 6 respectively
1! = 1, 2! = 2, 3! = 6, 4! = 24 (with base ending in 4 raised to an even power 24 giving unit digit 6)
2
Analyze factorial terms for k >= 5
Unit digit is 0 for all k >= 5
Factorials for k >= 5 contain factors 2 and 5, making the trailing digit 0
3
Sum the unit digits of the series sum
Sum of unit digits is 1 + 4 + 6 + 6 = 17, giving unit digit 7
Only terms from k = 1 to 4 contribute to the unit digit of the factorial sum
4
Find the unit digit of the power tower 7^(7^7) using cyclicity modulo 4
7^7 mod 4 = 3, so unit digit is 7^3 mod 10 = 3
The base 7 has a cyclicity of 4, and 7^7 is congruent to 3 modulo 4
5
Add the unit digits of both components
7 + 3 = 10, unit digit is 0
Combining the unit digit of the factorial sum (7) and the tower exponent term (3)

Key Concept

Combining factorial unit digit termination properties with exponent tower cyclicity modulo 4.
Question 94Question →

What is the final numerical value of the following expression evaluated using the standard BODMAS rule?

50−[18÷3×(4+2)]50 - [18 \div 3 \times (4 + 2)]
Show answer & explanation

Answer: 14

Answer

The final simplified value of the expression is 14.
Following the BODMAS order of operations: first simplify the inner bracket (4+2)=6(4 + 2) = 6. Next, evaluate inside the square bracket from left to right: 18÷3=618 \div 3 = 6, and then 6×6=366 \times 6 = 36. Finally, subtract from 50 to get 50−36=1450 - 36 = 14.

Step-by-Step Solution

1
Evaluate the innermost round brackets
4+2=64 + 2 = 6
According to BODMAS, operations inside brackets must be performed first.
2
Perform division inside the square brackets
18÷3=618 \div 3 = 6
Division and multiplication have equal priority and are performed from left to right.
3
Perform multiplication inside the square brackets
6×6=366 \times 6 = 36
Multiply the quotient obtained from division by the bracketed sum.
4
Subtract the result inside brackets from 50
50−36=1450 - 36 = 14
Perform subtraction as the final operation.

Key Concept

BODMAS / Order of Operations
Question 95Question →

What is the unit digit of the composite expression E=(1!+2!+3!+⋯+20!)2026+(31×32×33×⋯×320)E = (1! + 2! + 3! + \dots + 20!)^{2026} + (3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20})?

Show answer & explanation

Answer: 8

Answer

The unit digit of the expression is 8.
Evaluating the expression requires breaking it down into two components. First, for the factorial sum 1!+2!+3!+⋯+20!1! + 2! + 3! + \dots + 20!, every term from 5!5! onward contains factors of both 2 and 5, so its unit digit is 0. The unit digit of the sum is determined solely by 1!+2!+3!+4!=331! + 2! + 3! + 4! = 33, which has a unit digit of 3. Raising 3 to the power 2026 gives 320263^{2026}. Since the unit digits of powers of 3 repeat in cycles of 4 (3, 9, 7, 1) and 2026≡2(mod4)2026 \equiv 2 \pmod 4, the unit digit of 320263^{2026} is 32=93^2 = 9.

Second, the product 31×32×⋯×3203^1 \times 3^2 \times \dots \times 3^{20} simplifies using the exponent addition rule to 31+2+⋯+20=32103^{1+2+\dots+20} = 3^{210}. Dividing 210 by 4 leaves a remainder of 2, so 32103^{210} also has a unit digit of 32=93^2 = 9.

Adding the unit digits of both terms gives 9+9=189 + 9 = 18, resulting in a final unit digit of 8.

Step-by-Step Solution

1
Find the unit digit of the inner factorial sum S=1!+2!+3!+⋯+20!S = 1! + 2! + 3! + \dots + 20!.
The unit digit of SS is 3.
For all k≥5k \ge 5, k!k! is divisible by 10 and ends in 0. Thus, only the sum of the first four terms 1!+2!+3!+4!=1+2+6+24=331! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33 determines the unit digit.
2
Calculate the unit digit of the first term S2026≡32026S^{2026} \equiv 3^{2026}.
The unit digit of the first term is 9.
The unit digits of powers of 3 repeat in a cycle of 4 (3, 9, 7, 1). Dividing the exponent 2026 by 4 gives a remainder of 2 (2026=4×506+22026 = 4 \times 506 + 2). Therefore, the unit digit is 32=93^2 = 9.
3
Simplify the exponential product P=31×32×33×⋯×320P = 3^1 \times 3^2 \times 3^3 \times \dots \times 3^{20}.
The product simplifies to 32103^{210}.
By exponent multiplication rules, 31×32×⋯×320=3∑i=120i3^1 \times 3^2 \times \dots \times 3^{20} = 3^{\sum_{i=1}^{20} i}. The sum of the first 20 positive integers is 20×212=210\frac{20 \times 21}{2} = 210.
4
Calculate the unit digit of 32103^{210}.
The unit digit of the second term is 9.
Dividing the exponent 210 by 4 gives a remainder of 2 (210=4×52+2210 = 4 \times 52 + 2). Therefore, the unit digit is 32=93^2 = 9.
5
Combine the unit digits of the two terms.
The unit digit of EE is 8.
Adding the unit digits gives 9+9=189 + 9 = 18. The unit digit of 18 is 8.

Key Concept

Unit Digit and Cyclicity of Factorial and Exponential Expressions
Estimated Time:2m 0s
Question 96Question →
What exact numerical result is obtained when simplifying the following multi-bracket expression according to the BODMAS order of operations?
75% of 240−[38.5+{1512−(6.8×2.5−4.2+3.8‾)÷0.5}]75\% \text{ of } 240 - \left[ 38.5 + \left\{ 15\frac{1}{2} - \left( 6.8 \times 2.5 - \overline{4.2 + 3.8} \right) \div 0.5 \right\} \right]
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Answer: 144

Answer

The final simplified numerical value of the given expression is 144.
Following the BODMAS rule strictly: first evaluate the vinculum 4.2+3.8‾=8\overline{4.2 + 3.8} = 8; then resolve round brackets (6.8×2.5−8=9)(6.8 \times 2.5 - 8 = 9); next resolve curly brackets (15.5−9÷0.5=−2.5)(15.5 - 9 \div 0.5 = -2.5); then square brackets (38.5−2.5=36)(38.5 - 2.5 = 36); and finally compute 75% of 240=18075\% \text{ of } 240 = 180. Subtracting 3636 from 180180 gives the correct result of 144.

Step-by-Step Solution

1
Evaluate the vinculum (bar) operator first
\overline{4.2 + 3.8} = 8
According to BODMAS, the vinculum takes precedence over round brackets.
2
Simplify the terms inside the round brackets (parentheses)
6.8 \times 2.5 - 8 = 17 - 8 = 9
Multiplication precedes subtraction inside the innermost round brackets.
3
Simplify the terms inside the curly brackets (braces)
15.5 - (9 \div 0.5) = 15.5 - 18 = -2.5
Division (9÷0.5=189 \div 0.5 = 18) is performed before subtracting from 15.515.5 (151215\frac{1}{2}).
4
Simplify the terms inside the square brackets
38.5 + (-2.5) = 36
Adding the negative result from the curly brackets to 38.538.5 yields 3636.
5
Calculate the percentage 'Of' operation and perform final subtraction
75\% \text{ of } 240 - 36 = 180 - 36 = 144
'Of' operation (75%×240=18075\% \times 240 = 180) is calculated before subtracting the bracketed total.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets
Estimated Time:2m 0s
Question 97Question →
What is the result when the mathematical expression 45−[12+(8÷2×3)]45 - [12 + (8 \div 2 \times 3)] is simplified strictly adhering to the standard BODMAS order of operations?
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Answer: 21

Answer

21
Following the BODMAS rule, operations inside the innermost parentheses are solved first from left to right: 8÷2=48 \div 2 = 4, then 4×3=124 \times 3 = 12. Next, addition inside the square brackets gives 12+12=2412 + 12 = 24. Finally, subtracting 24 from 45 yields 21.

Step-by-Step Solution

1
Evaluate division inside the parentheses
8 ÷ 2 = 4
Division and multiplication have equal priority and are evaluated from left to right inside parentheses.
2
Evaluate multiplication inside the parentheses
4 × 3 = 12
Completing the operations within the parentheses.
3
Add the values inside the square brackets
12 + 12 = 24
Evaluating the bracketed addition.
4
Subtract from the outer term
45 - 24 = 21
Performing final subtraction outside the brackets.

Key Concept

BODMAS Rule (Order of Operations)
Estimated Time:45s
Question 98Question →

Determine the unit digit of the numerical expression N=(333444×444333)+(777888×888777)−222555N = (333^{444} \times 444^{333}) + (777^{888} \times 888^{777}) - 222^{555}.

Show answer & explanation

Answer: 4

Answer

The unit digit of the given expression is 4.
By analyzing cyclicity of each base (3,4,7,8,23, 4, 7, 8, 2), we find that 333444→1333^{444} \rightarrow 1, 444333→4444^{333} \rightarrow 4, 777888→1777^{888} \rightarrow 1, 888777→8888^{777} \rightarrow 8, and 222555→8222^{555} \rightarrow 8. Thus, the overall unit digit is (1×4)+(1×8)−8=4+8−8=4(1 \times 4) + (1 \times 8) - 8 = 4 + 8 - 8 = 4.

Step-by-Step Solution

1
Calculate unit digit of 333444×444333333^{444} \times 444^{333}
Unit digit is 4
Base 3 has cyclicity 4; 444(mod4)=0  ⟹  34→1444 \pmod 4 = 0 \implies 3^4 \rightarrow 1. Base 4 has cyclicity 2; odd exponent 333  ⟹  41→4333 \implies 4^1 \rightarrow 4. Total unit digit for term 1 = 1×4=41 \times 4 = 4.
2
Calculate unit digit of 777888×888777777^{888} \times 888^{777}
Unit digit is 8
Base 7 has cyclicity 4; 888(mod4)=0  ⟹  74→1888 \pmod 4 = 0 \implies 7^4 \rightarrow 1. Base 8 has cyclicity 4; 777(mod4)=1  ⟹  81→8777 \pmod 4 = 1 \implies 8^1 \rightarrow 8. Total unit digit for term 2 = 1×8=81 \times 8 = 8.
3
Calculate unit digit of 222555222^{555}
Unit digit is 8
Base 2 has cyclicity 4; 555(mod4)=3  ⟹  23→8555 \pmod 4 = 3 \implies 2^3 \rightarrow 8.
4
Evaluate combined expression unit digit
4
Combine term unit digits: (4+8−8)=4(4 + 8 - 8) = 4.

Key Concept

Unit Digit and Cyclicity of Numbers
Question 99Question →

What is the unit digit of the numerical expression N=(137137×264102)+619199N = (137^{137} \times 264^{102}) + 619^{199}?

Show answer & explanation

Answer: 1

Answer

1
The unit digit of 137137137^{137} is derived from 137(mod4)=1137 \pmod 4 = 1, giving 71=77^1 = 7. The unit digit of 264102264^{102} is 6 because the exponent 102 is even. Multiplying these unit digits gives 7×6=427 \times 6 = 42, which contributes a unit digit of 2. The unit digit of 619199619^{199} is 9 because 199 is odd. Adding the unit digits yields 2+9=112 + 9 = 11, whose unit digit is 1.

Step-by-Step Solution

1
Determine the unit digit of 137137137^{137} using the cyclicity of 7.
The exponent 137 leaves a remainder of 1 when divided by 4, giving a unit digit of 71=77^1 = 7.
The unit digit pattern for powers of 7 repeats every 4 powers (7, 9, 3, 1).
2
Determine the unit digit of 264102264^{102} using the cyclicity of 4.
Since 102 is an even exponent, 4even4^{\text{even}} yields a unit digit of 6.
Powers of 4 end in 4 for odd exponents and 6 for even exponents.
3
Calculate the unit digit of the product (137137×264102)(137^{137} \times 264^{102}).
(7×6)=42(7 \times 6) = 42, which has a unit digit of 2.
The unit digit of a product depends only on the product of the unit digits of its factors.
4
Determine the unit digit of 619199619^{199} using the cyclicity of 9.
Since 199 is an odd exponent, 9odd9^{\text{odd}} yields a unit digit of 9.
Powers of 9 end in 9 for odd exponents and 1 for even exponents.
5
Sum the unit digits to find the final unit digit of expression NN.
2+9=112 + 9 = 11, which has a unit digit of 1.
The unit digit of the sum of two terms is the unit digit of the sum of their individual unit digits.

Key Concept

Unit digit determination using exponent cyclicity rules for base digits 4, 7, and 9.
Estimated Time:1m 15s
Question 100Question →

If M=7+210+7−210M = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} and xx satisfies the exponential equation (43)2x−1=(116)x−4\left(\sqrt[3]{4}\right)^{2x-1} = \left(\frac{1}{16}\right)^{x-4} with (43)2x−1=2k\left(\sqrt[3]{4}\right)^{2x-1} = 2^k, what is the value of M2+kM^2 + k?

Show answer & explanation

Answer: 23.5

Answer

The value of M2+kM^2 + k is 23.5.
By writing 7±2107 \pm 2\sqrt{10} as (5±2)2(\sqrt{5} \pm \sqrt{2})^2, the radical simplifies cleanly to M=25M = 2\sqrt{5}, giving M2=20M^2 = 20. Rewriting the index equation in terms of base 2 yields 4x−23=−4x+16\frac{4x-2}{3} = -4x + 16, which gives x=258x = \frac{25}{8} and exponent k=3.5k = 3.5. Adding M2M^2 and kk results in 23.5.

Step-by-Step Solution

1
Simplify the nested surd expression for M
M = 2\sqrt{5}, so M^2 = 20
Recognize that 7±210=(5±2)27 \pm 2\sqrt{10} = (\sqrt{5} \pm \sqrt{2})^2.
2
Convert both sides of the exponential equation to base 2
24x−23=2−4x+162^{\frac{4x-2}{3}} = 2^{-4x+16}
Apply laws of indices: 43=22/3\sqrt[3]{4} = 2^{2/3} and 116=2−4\frac{1}{16} = 2^{-4}.
3
Solve for x by equating the powers of 2
x=258=3.125x = \frac{25}{8} = 3.125
Since bases are equal, the powers must be equal.
4
Determine the exponent value k
k = 3.5
Substitute x into the exponent expression k=−4x+16k = -4x + 16.
5
Calculate the final combined expression M^2 + k
23.5
Add M2=20M^2 = 20 and k=3.5k = 3.5.

Key Concept

Nested radical simplification using binomial square expansion combined with solving exponential equations via prime base unification.
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