Elementary Number Theory MCQ

Explanation: This question asks us to determine the greatest common divisor (G.C.D.) of two numbers expressed as products of prime factors. The aim is to identify the highest common factor shared by both numbers based on their prime decomposition.

To solve such problems, each number is written as a product of prime factors. The G.C.D. is found by taking only those prime factors that are common to both numbers, and for each such factor, selecting the smallest power (or frequency) with which it appears.

In the given expressions, both numbers share some common primes. By comparing their occurrences, we select only the common ones with the lowest exponent. This ensures that the divisor divides both numbers completely without exceeding their shared factor limit.

For example, if two numbers have 2 × 2 × 3 and 2 × 3 × 3, then only one 2 and one 3 are common in both, so we take those. The same logic applies here.

This method guarantees finding the greatest number that divides both values exactly by focusing on shared prime factors and their minimum occurrences.

Explanation: This problem involves finding the greatest common divisor (G.C.D.) of three numbers given in their prime factorized forms. The goal is to identify the largest number that divides all three numbers exactly.

To approach this, we compare the prime factors of all three numbers. Only those primes that are present in every number are considered. For each such common prime, we select the lowest number of times it appears among the numbers.

Each expression contains different combinations of prime numbers. By carefully identifying which primes are common to all three, we eliminate any that do not appear in every expression. Then, among the shared primes, we consider their minimum occurrence.

For instance, if one number contains 2 twice and another only once, we take it only once. This ensures the divisor is common to all.

This systematic comparison helps us find the largest factor that divides all given numbers exactly by focusing only on shared primes with the smallest powers.

Explanation: This question requires finding the greatest number that divides two given numbers such that specific remainders are left in each case. The idea is to adjust the given numbers so that they become exactly divisible.

When a number leaves a remainder after division, subtracting that remainder makes it perfectly divisible. So, we first reduce both numbers by their respective remainders. This transforms the problem into finding a common divisor of the adjusted values.

After adjustment, the task becomes finding the G.C.D. of the two new numbers. The G.C.D. represents the largest number that divides both values exactly, satisfying the required condition.

For example, if a number leaves remainder 3 when divided, subtracting 3 gives a multiple of the divisor. Applying this logic to both numbers ensures consistency.

Thus, by converting the problem into a standard G.C.D. calculation, we identify the required largest number that satisfies the remainder conditions.

Explanation: This problem is based on the basic division algorithm, which connects dividend, divisor, quotient, and remainder in a fixed relationship. The goal is to reconstruct the original number using the given components.

According to the division rule, any number can be expressed as: dividend = (divisor × quotient) + remainder. This formula allows us to calculate the original number when the other three values are known.

Here, the divisor, quotient, and remainder are provided. By substituting these values into the formula, we obtain the required number.

For instance, if a number divided by 3 gives quotient 4 and remainder 1, then the number equals (3 × 4) + 1. The same logic applies here.

This approach ensures an exact reconstruction of the number based on the given division conditions, following the fundamental division identity.

Explanation: This question uses the Euclidean algorithm, where successive division steps generate quotients. These quotients help reconstruct the original pair of numbers when the G.C.D. is known.

The process begins with the last quotient and works backward. Each step involves expressing the previous number as a combination of the current divisor and remainder. By reversing the division steps, we gradually rebuild the original numbers.

The given quotients guide how many times one number fits into another at each stage. Starting from the G.C.D., we iteratively apply the reverse process using the quotients in reverse order.

For example, if the last step gives a remainder equal to the G.C.D., earlier steps can be reconstructed using multiplication and addition based on quotients.

This systematic backward reconstruction ultimately yields the original pair of numbers consistent with the given quotients and G.C.D.

Explanation: This problem also relies on the Euclidean algorithm, where quotients obtained during division help reconstruct the original numbers when the G.C.D. is known.

Starting from the G.C.D., we reverse the division steps using the quotients provided. Each quotient indicates how many times one number divides another at a particular stage. By working backward, we form equations that rebuild the numbers step by step.

The process involves repeated substitution, where each stage depends on the previous one. This creates a chain of relationships that eventually leads to the original pair.

Once both numbers are reconstructed, their sum can be determined. The method ensures consistency with both the G.C.D. and the sequence of quotients.

This reverse approach transforms division steps into a constructive process for recovering the original numbers.

Explanation: This question involves finding the least common multiple (LCM) of two numbers expressed in terms of prime factors. The LCM is the smallest number divisible by both given numbers.

To compute the LCM using prime factorization, we consider all distinct primes present in both numbers. For each prime, we select the highest power with which it appears in either number.

In the given expressions, both X and Y share some primes while also having distinct ones. For shared primes, we compare their powers and take the maximum. For primes appearing in only one expression, we include them as they are.

For example, if one number has 2³ and another has 2², the LCM includes 2³. This ensures divisibility by both.

This method guarantees the smallest number that contains all required prime factors in sufficient quantity to be divisible by both numbers.

Explanation: This problem involves finding a number that satisfies multiple remainder conditions simultaneously. The idea is to convert these conditions into a form suitable for applying LCM concepts.

In each case, the remainder is less than the divisor but follows a pattern. Subtracting the remainder from the number makes it divisible by the divisor. Observing carefully, the differences between divisor and remainder are consistent.

By adjusting the number accordingly, we reduce the problem to finding a common multiple of the divisors. Specifically, we look for a number that, when modified, becomes divisible by all given numbers.

This leads us to compute the LCM of the divisors. After finding the LCM, we adjust back using the remainder pattern.

This approach ensures that the number satisfies all given conditions simultaneously in the smallest possible way.

Explanation: This question asks for the smallest number that leaves the same remainder when divided by several numbers. Such problems are solved using LCM principles.

If a number leaves remainder 1 when divided by several divisors, subtracting 1 makes it divisible by all those divisors. This transforms the problem into finding a common multiple.

We then compute the LCM of the given divisors, which gives the smallest number divisible by all of them. After obtaining this value, we add back the remainder to get the required number.

For example, if a number leaves remainder 2 when divided by several numbers, subtracting 2 makes it divisible by all.

This method simplifies the problem by converting it into a standard LCM calculation followed by a simple adjustment.

Explanation: This problem asks for the smallest number that is divisible by all integers from 1 to 10. Such a number must be a common multiple of all these integers.

The smallest such number is found by calculating the LCM of the numbers from 1 to 10. To do this efficiently, we express each number in terms of prime factors and take the highest power of each prime required.

For instance, numbers like 8 contribute higher powers of 2, while 9 contributes a higher power of 3. We ensure that all such requirements are included.

By combining these highest powers, we construct the smallest number divisible by all given integers.

This ensures that the number leaves no remainder when divided by any number from 1 to 10, satisfying the condition completely.

Explanation: This problem involves multiple remainder conditions with different divisors. The strategy is to convert these into a common divisibility condition.

Each remainder is slightly less than its divisor. Subtracting the remainder from the divisor reveals a pattern. By adjusting the number appropriately, we can make it divisible by each divisor.

Thus, we transform the problem into finding a number that becomes divisible by 18, 24, and 30 after a fixed adjustment. This leads us to calculate the LCM of these divisors.

Once the LCM is obtained, we reverse the adjustment to get the required number.

This method ensures that all remainder conditions are satisfied simultaneously in the smallest possible number.

Explanation: This question is based on the fundamental relationship between two numbers, their G.C.D., and their L.C.M. The product of the two numbers equals the product of their G.C.D. and L.C.M.

Using this relationship, we can form an equation involving the known values. Since one number is given, we substitute it into the equation and solve for the other number.

This avoids the need for factorization and provides a direct method for finding the missing number.

For example, if the product of two numbers is known and one number is given, the other can be found by division.

This principle provides a quick and reliable way to determine the unknown number using the given G.C.D. and L.C.M. values.

Explanation: This question uses the fundamental relationship between two numbers, their G.C.D., and their L.C.M. The product of two numbers is always equal to the product of their G.C.D. and L.C.M.

To solve this, we use the formula: Product of numbers = G.C.D. × L.C.M. Since the product and L.C.M. are given, we can rearrange the formula to find the G.C.D. by dividing the product by the L.C.M.

This method avoids lengthy factorization and provides a direct calculation. It works because the G.C.D. accounts for common factors, while the L.C.M. accounts for the highest powers of all primes involved.

For example, if two numbers multiply to a known value and their L.C.M. is known, dividing gives the G.C.D.

Thus, using this identity, we can efficiently determine the required value by simple division.

Explanation: This problem again uses the relationship between G.C.D., L.C.M., and the product of two numbers. The formula states that Product = G.C.D. × L.C.M.

Here, both numbers and their L.C.M. are given. First, we compute the product of the two numbers. Then, using the formula, we divide the product by the L.C.M. to obtain the G.C.D.

This approach is efficient and avoids detailed prime factorization. It relies on the fundamental property that links these three quantities.

For instance, if two numbers are known along with their L.C.M., the G.C.D. can always be found using this formula.

This ensures an accurate result by directly applying the relationship between these values.

Explanation: This question tests the relationship between two numbers and their L.C.M. In some cases, the product of two numbers equals their L.C.M., but this is not always true.

The key concept is that when two numbers are relatively prime (i.e., their G.C.D. is 1), their product equals their L.C.M. However, if they share common factors, the L.C.M. is smaller than their product.

Thus, we analyze different types of number pairs. For relatively prime numbers, the condition holds. For numbers with common factors, the product includes repeated factors, while the L.C.M. avoids duplication.

For example, numbers like 6 and 8 share a common factor, so their product and L.C.M. differ.

By understanding this distinction, we can identify which pairs do not satisfy the condition.

Explanation: This question is about properties of relations in mathematics, specifically the relation “is a factor of.” Common properties include reflexive, symmetric, transitive, and antisymmetric.

A relation is reflexive if every element is related to itself, which holds true here since every number divides itself. It is transitive if a relation passes through intermediate elements, which also applies.

However, symmetry requires that if one number is related to another, the reverse must also be true. In the case of factors, if a divides b, it does not imply that b divides a.

Antisymmetry means that if both a divides b and b divides a, then a equals b, which holds true.

Thus, by analyzing these properties, we determine which one is not satisfied.

Explanation: This question refers to a fundamental concept in number theory regarding the representation of integers. The theorem states that every integer greater than 1 can be expressed as a product of prime numbers in a unique way.

This uniqueness means that, apart from the order of factors, no other combination of primes can represent the same number. This property is essential for many arithmetic operations, including finding G.C.D. and L.C.M.

The theorem forms the foundation for prime factorization methods used in solving various mathematical problems.

For example, the number 60 can be written as 2 × 2 × 3 × 5, and no other SET of primes will give the same result.

This concept is widely recognized under a specific name in mathematics due to its importance.

Explanation: This problem involves comparing two expressions of the same number written as products of prime factors. Since both expressions represent the same value, their prime factorizations must match.

According to the uniqueness of prime factorization, the SET of primes and their powers must be identical in both expressions. This allows us to equate corresponding prime factors.

By comparing both sides, we identify which primes correspond to each other. Since p, q, r, and s are distinct primes, each must match with a specific prime on the other side.

Once the matching is done, we determine the values of these primes and compute their sum.

This method relies on the fundamental principle that a number has only one unique prime factorization.

Explanation: This question evaluates understanding of properties related to prime numbers, relatively prime numbers, and divisibility rules.

Prime numbers are numbers greater than 1 with only two factors. Relatively prime numbers are pairs of numbers whose G.C.D. is 1, even if they are not prime themselves.

Some statements describe true properties, such as the relationship between G.C.D. and L.C.M. for relatively prime numbers. Others involve divisibility rules that must hold under all conditions.

To identify the false statement, each option must be checked against known mathematical principles. Any statement that contradicts these rules is incorrect.

For example, a claim about divisibility that does not hold universally would be false.

By carefully evaluating each statement, we can determine which one does not align with established concepts.