Number Theory

Step 1: Concept
Solving linear equations.
Step 2: Analysis
3x = 20 - 5
3x = 15
Step 3: Conclusion
x = 15 / 3 = 5.
Final Answer: (C)

Step 1: Understand the Concept
To find the total number of divisors (factors) of any composite number , we first express the number as a product of its prime factors (prime factorization). If the prime factorization is , then the total number of divisors is calculated using the formula:
Total Divisors =
This formula accounts for all possible combinations of the prime factors, including the number 1 and the number itself.

Step 2: Analysis of the Number 360
First, we perform the prime factorization of 360:
.
Here, the exponents are , , and .
Applying the formula to find the total divisors:
Total Divisors =
Total Divisors = .

Step 3: Conclusion & Final Adjustment
The total count of 24 divisors includes every factor from 1 to 360. However, the question specifically asks for the number of divisors excluding 1 and 360. Therefore, we must subtract these 2 specific divisors from our total count:
Required number of divisors = .
Thus, there are 22 divisors of 360 when excluding the number 1 and the number itself.

Final Answer: (A)

Step 1: Understand the Concept
This problem is based on the principle of Time and Work, specifically involving Inverse Proportion. In such scenarios, the number of men working is inversely proportional to the time taken to complete the work (assuming the total work remains constant). This means that if you increase the number of workers, the time required to finish the task will decrease.

Step 2: Analysis & Application
We use the work-equivalence formula: , where represents the number of men and represents the number of days.
From the question, we have the initial data:
and .
The new condition is:
and we need to find .
Substituting these values into the formula:
.

Step 3: Calculation & Conclusion
First, calculate the total "man-days" required for the project:
.
Now, solve for by dividing both sides by 10:
.
Therefore, with the workforce doubled, the time taken to build the wall is halved, resulting in 6 days.

Final Answer: (B)

Step 1: Understand the Concept
To solve problems involving a train crossing an object with its own length (like a bridge, platform, or tunnel), we use the basic formula: Speed = Distance / Time. The key point here is that the Total Distance covered by the train to completely clear the bridge is the sum of the length of the train and the length of the bridge itself.

Step 2: Analysis & Unit Conversion
First, we must ensure all units are consistent. Since the length and time are given in meters and seconds, we convert the speed from km/hr to m/s:
Speed = .

Next, we calculate the total distance covered in 10 seconds at this speed:
Total Distance = .

Since the Total Distance is the sum of the train and bridge lengths, we find the bridge length by subtracting the train's length from the total:
Length of Bridge = .

Step 3: Conclusion
The calculation ( ) results in 100 meters. Therefore, the length of the bridge is 100m.

Final Answer: (A)

Step 1: Understand the Concept
This problem is based on the core principles of Profit and Loss. When an item is sold at a profit, the Selling Price (SP) is equal to the original Cost Price (CP) plus the profit earned. Because profit is always calculated as a percentage of the Cost Price, making a 10% profit means the final Selling Price is exactly 110% of the Cost Price.

Step 2: Identify Given Information & Analyze
Based on the problem, we have the following values:
Selling Price (SP) = Rs. 440
Profit Margin = 10%
Applying our concept, we can set up the relationship: SP = 110% of CP.
Written mathematically, this becomes:


Step 3: Calculation & Conclusion
To find the actual Cost Price, we simply rearrange the equation to solve for CP:

By dividing 440 by 110, we get 4. Multiplying that by 100 gives us our final value:

Therefore, the original Cost Price of the item was Rs. 400.

Final Answer: (A)

Step 1: Understand the Concept
When a boat travels downstream, it moves in the exact same direction as the flow of the river. Because the water's current is actively pushing the boat forward, we must add the speed of the stream to the boat's normal speed in still water to calculate its total effective speed.
Formula: Speed in Downstream = Speed in Still Water + Speed of Stream.

Step 2: Identify Given Information & Analyze
Based on the question, we are given the following values:
Speed of the boat in still water = 12 km/hr
Speed of the stream = 4 km/hr
Substituting these values into our downstream formula, we get:
Downstream Speed = 12 + 4.

Step 3: Calculation & Conclusion
By simply adding the two speeds together (12 + 4), we find that the effective speed of the boat as it travels with the current is 16 km/hr.

Final Answer: (B)

Step 1: Concept
Average = Sum of values / Number of values.
Step 2: Analysis
Multiples: 3, 6, 9, 12, 15. Sum = 45.
Step 3: Conclusion
Average = 45 / 5 = 9.
Final Answer: (B)

Step 1: Concept
If , total divisors .
Step 2: Analysis
.
Total divisors .
Step 3: Conclusion
Excluding 1 and 360, the number of divisors is .
Final Answer: (A)

Step 1: Use Euler's Theorem.
We are asked to find the remainder when is divided by 5. By Euler's theorem, if and are coprime (i.e., ), then: where is Euler's totient function.

Step 2: Compute .
Since 5 is a prime number, . Therefore, by Euler's theorem:

Step 3: Simplify the exponent.
We need to find . We can break the exponent down using the fact that . First, express 30 as: So, Since , we have: Thus,

Step 4: Calculate .
Now, calculate : So,

Step 5: Conclusion.
Thus, , so the least remainder when is divided by 5 is , corresponding to option (C).

Step 1: Understand the operation in the group.
We are given the group , which represents a group under multiplication modulo 5. This means the operations are performed modulo 5.

Step 2: Express the equation in terms of the group operation.
The equation given is: First, solve for , the multiplicative inverse of 2 modulo 5. By checking multiples of 2 modulo 5, we find: So, modulo 5.

Step 3: Substitute the inverse into the equation.
Now substitute into the equation: Simplifying:

Step 4: Reduce modulo 5.
Since we are working modulo 5, reduce the terms: So the equation becomes:

Step 5: Solve for .
Divide both sides by 4 (which is invertible modulo 5): Since for any , the solution is .

Step 6: Conclusion.
Thus, the solution to the equation is , corresponding to option (A).

Step 1: Simplify the terms.
We are given the expression: Start by simplifying the third term by rationalizing the denominator.

Step 2: Rationalize the denominator.
To rationalize , multiply the numerator and denominator by : Simplifying the denominator: Thus:

Step 3: Combine the terms.
Now, combine all terms:

Step 4: Conclusion.
After simplifying further, the final expression evaluates to , corresponding to option (D).

Step 1: Understand logarithms.
Logarithms with different bases behave differently. We are given four different logarithmic functions: , , , and .

Step 2: Compare the values for different logarithmic bases.
To determine the correct order, recall that: - increases faster than , which increases faster than , which in turn increases faster than when . - Therefore, for .

Step 3: Conclusion.
Thus, the correct increasing order is , corresponding to option (B).

Step 1: Expand the given expression.
We are given and , and need to calculate the value of . First, calculate , , and :

Step 2: Substitute into the expression.
Now substitute these into the original expression:
Simplifying:

Step 3: Final simplification.
Now, factor out from the expression: Thus, the correct answer is , which corresponds to option (A).

Step 1: Break down the square root expression.
We are given:
First, simplify the square roots: Thus, the expression becomes:

Step 2: Square both sides.
Let’s square the expression: Thus the expression simplifies to:

Step 1: Convert logarithms to the same base.
We are given the equation:

We can convert to base 2:

Thus, the equation becomes:


Step 2: Eliminate the fractions.
Multiply both sides of the equation by 2 to eliminate the fraction:

Step 3: Apply logarithmic properties.
Using the logarithmic property , we can rewrite the equation as:

Step 4: Solve the equation.
Since the logarithms are now equal, the arguments must be equal as well: Expanding the square:

Step 5: Rearrange the equation.
Rearrange the equation to bring all terms to one side: Factor the quadratic:

Step 6: Find the solutions.
Thus, the solutions are: We must check if these solutions satisfy the original logarithmic equation. For : For :

Step 7: Conclusion.
Thus, the only valid solution is , so the number of solutions is 1.

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
A graph where every edge has a direction is called a directed graph or digraph.
Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



Always false contradiction.
Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
(by commutativity)

Step 3: Final Answer

Step 1: Use magnitude of cross product}
Given , so Similarly, from , Step 2: Use perpendicularity}
Since cross product gives a vector perpendicular to both vectors, Thus, angles are , so Step 3: Solve equations}
Substitute into second equation: Step 4: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
Point lies between the lines if:



Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}

Given


Step 3: Final Answer

Step 1: Understanding the Concept:
represents the set of all multiples of . Intersection of sets of multiples gives the set of multiples of the LCM.

Step 2: Detailed Explanation:
= set of multiples of , i.e., . = set of multiples of . Their intersection is the set of numbers that are multiples of both and , i.e., multiples of . Since and are coprime, . Thus, . Given that , we have , which implies .

Step 3: Final Answer:
, which corresponds to option (C).

Step 1: , . So .}
Step 2: Multiply by inverse of 3 mod 5 (which is 2): . So .}
Step 3: Among options, ? Actually , not 4. , so is correct.}

Step 1: , so .}
Step 2: , so .}
Step 3: Remainder is 4. Option (C).}

Step 1: Euclidean algorithm: , , , , .}
Step 2: GCD = .}

Step 1: Understanding the Concept:
means .
Step 2: Detailed Explanation:
must be divisible by 4.
.
For : . Verify: , ; . \checkmark
Step 3: Final Answer:
One value of is .

Step 1: Understanding the Concept:
means is the greatest common divisor of and .
Step 2: Detailed Explanation:
By definition of GCD, divides both and . So both and are integers, i.e.\ and .
Option (A) is therefore correct.
Step 3: Final Answer:
is true.

Step 1: Understanding the Concept:
Express each factor modulo 124 and then multiply the residues.
Step 2: Detailed Explanation:
, , .
Product .
Step 3: Final Answer:
The least positive remainder is .

Step 1: Understanding the Concept:
For , is divisible by 14.
Step 2: Detailed Explanation:

For , contains both 2 and 7, so divisible by 14.
So only need sum
=
: , remainder 5
Step 3: Final Answer:
Remainder = 5.

Step 1: Understanding the Concept:
In a group, multiply both sides on right by .
Step 2: Detailed Explanation:

Multiply right by :

So unique solution.
Step 3: Final Answer:
Unique solution .

Step 1: Understanding the Concept:
Tautology is always true regardless of truth values.
Step 2: Detailed Explanation:
(A):
(B):
(C):
(D): , which is not always true.
Step 3: Final Answer:
(D) is not a tautology.

Step 1: Understanding the Concept:
Divide by 2 repeatedly.
Step 2: Detailed Explanation:
60 ÷ 2 = 30 remainder 0
30 ÷ 2 = 15 remainder 0
15 ÷ 2 = 7 remainder 1
7 ÷ 2 = 3 remainder 1
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read backwards:
Step 3: Final Answer:
111100.

Step 1: Understanding the Concept:
Convert binary to decimal: each digit .
Step 2: Detailed Explanation:

=
Step 3: Final Answer:
19.50.

Step 1: Understanding the Concept:
Sum of squares of first n natural numbers.
Step 2: Detailed Explanation:
is sum of squares, which is integer.
So is divisible by 6.
Step 3: Final Answer:
Integral multiple of 6.

Concept:

Step 1: Let.


Step 2: Compute.

Concept: Let .

Step 1: Substitute.
Multiply by :

Step 2: Find required expression.
Using , reduce powers: Simplifying gives:

Concept: Convert all terms to base 3.

Step 1: Rewrite bases.


Step 2: Convert expressions.


Step 3: Equate powers.
Solving consistently gives:

Concept: Use identity:

Step 1: Rewrite.


Step 2: Substitute.


Step 3: Final.

Concept: To find last two digits, compute .

Step 1: Find cycle:

Step 2: Since :

Step 3: Therefore, last two digits = 01.

Concept:

Step 1: Prime factorization:

Step 2: Conditions:
•Even at least one factor of
•Divisible by 15 must include and

Step 3: Choices:
• 3 ways
• 3 ways
• 3 ways
• 2 ways
• 2 ways

Step 4:

Concept: Convert ratios into actual values.

Step 1:

Step 2: Match : So:

Step 3: