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Fundamental Theorem of Arithmetic — Class 10 CBSE Mathematics MCQs with Solutions
Free Class 10 CBSE Mathematics Fundamental Theorem of Arithmetic MCQs with step-by-step solutions (33 questions). Part of Real Numbers. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Fundamental Theorem of Arithmetic · easy · theory
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a **unique** way (ignoring order). What does 'unique' mean here?
A. There is only one way to write the number as a product of primes (disregarding the order of factors) ✓ Correct
B. Each number can be prime or composite, but not both
C. All composite numbers have different prime factors
D. The number of prime factors is always different for each number
Solution: Uniqueness of prime factorization means that if you factor a number into primes, you will always get the same set of primes with the same frequencies, no matter which method you use. For example, $12 = 2 \times 2 \times 3 = 2^2 \times 3$ is the *only* way to express 12 as a product of primes. You cannot express 12 as $2 \times 5 + \text{something}$ or any other prime combination.
Q2 — Fundamental Theorem of Arithmetic · easy · theory
According to the Fundamental Theorem of Arithmetic, what can we say about any integer greater than 1?
A. It is either prime or can be uniquely expressed as a product of prime numbers ✓ Correct
B. It is always composite
C. It must be even or odd, never both
D. It can be expressed as a sum of primes in multiple ways
Solution: The Fundamental Theorem of Arithmetic covers all integers greater than 1. A number is either: (1) prime (has no prime factorization other than itself), or (2) composite (can be uniquely factorized into primes). This theorem guarantees that every number falls into one of these two categories, and composite numbers have a unique factorization.
Q3 — Fundamental Theorem of Arithmetic · medium · theory
Two numbers $a$ and $b$ have prime factorizations: $a = 2^3 \times 3^2 \times 5$ and $b = 2^2 \times 3 \times 7$. Based on the Fundamental Theorem of Arithmetic, which reasoning is valid?
A. $a$ and $b$ are different numbers because their prime factorizations are different ✓ Correct
B. $a$ and $b$ might be the same number with different factorizations
C. One must be prime and the other composite
D. We cannot determine if they are equal without knowing their numerical values
Solution: The uniqueness part of the Fundamental Theorem tells us that if two numbers have different prime factorizations, they must be different numbers. Since $a$'s factorization includes $5$ but $b$'s includes $7$, they cannot be equal. This reasoning is powerful because it lets us compare numbers algebraically without computing their actual values.
Q4 — Fundamental Theorem of Arithmetic · hard · theory
Suppose someone claims: 'I found a new way to factor the number 60 that is different from $60 = 2^2 \times 3 \times 5$.' If the Fundamental Theorem of Arithmetic is true, what must be wrong with this claim?
A. The claimed factorization must be incorrect or not use only prime factors ✓ Correct
B. The number 60 is actually prime, so it has no other factorization
C. The FTA only applies to numbers greater than 60
D. The person made a computational error but the alternate factorization could still exist
Solution: The Fundamental Theorem of Arithmetic guarantees that there is *only one* way to express 60 as a product of primes. If someone claims to have found a different factorization, either (1) their factorization is wrong, (2) it includes composite factors (not just primes), or (3) it is the same factorization written in a different order. The theorem rules out any genuinely different factorization.
Q5 — Fundamental Theorem of Arithmetic · hard · theory
The Fundamental Theorem of Arithmetic states that prime factorization is unique. What does this uniqueness logically imply about the divisibility properties of a number?
A. All possible divisors of a number can be determined from its prime factorization, and no other divisors exist ✓ Correct
B. A number can have divisors that are not made from its prime factors
C. Divisibility is a matter of opinion depending on the method used
D. Every number has infinitely many divisors
Solution: Since the prime factorization is unique, the set of all divisors of a number is completely determined by its primes. For $12 = 2^2 \times 3$, the divisors are exactly those formed by taking powers of 2 from $0$ to $2$ and powers of 3 from $0$ to $1$: {1, 2, 3, 4, 6, 12}. Uniqueness means there are no hidden divisors outside this system. This makes the divisibility structure predictable and complete.
Q6 — Fundamental Theorem of Arithmetic · medium · theory
If the HCF of 65 and 117 is expressible in the form 65m - 117, then the value of m is:
A. 4
B. 2 ✓ Correct
C. 1
D. 3
Q7 — Fundamental Theorem of Arithmetic · medium · theory
If two positive integers a and b are written as a = x³y² and b = xy³, where x, y are prime numbers, then HCF(a, b) is:
A. xy
B. xy² ✓ Correct
C. x³y³
D. x²y²
Q8 — Fundamental Theorem of Arithmetic · medium · theory
The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is:
A. 10
B. 100
C. 504
D. 2520 ✓ Correct
Q9 — Fundamental Theorem of Arithmetic · easy · theory
The sum of the exponents of the prime factors in the prime factorization of 196 is:
A. 1
B. 2
C. 4 ✓ Correct
D. 6
Q10 — Fundamental Theorem of Arithmetic · medium · theory
If n is any natural number, then 6ⁿ - 5ⁿ always ends with:
A. 1 ✓ Correct
B. 3
C. 5
D. 7
Q11 — Fundamental Theorem of Arithmetic · easy · theory
If HCF(a, b) = 12 and a · b = 1800, then LCM(a, b) is:
A. 3600
B. 150 ✓ Correct
C. 900
D. 120
Q12 — Fundamental Theorem of Arithmetic · medium · theory
If m^n = 32, where m and n are positive integers, then the value of n^(mn) is:
A. 5²
B. 5¹⁰ ✓ Correct
C. 5²⁵
D. 2²⁵
Q13 — Fundamental Theorem of Arithmetic · medium · theory
If p₁ and p₂ are two odd prime numbers such that p₁ > p₂, then p₁² - p₂² is:
A. an even number ✓ Correct
B. an odd number
C. an odd prime number
D. a prime number
Q14 — Fundamental Theorem of Arithmetic · medium · theory
For any positive integer n, n² - n is divisible by:
A. 3
B. 2 ✓ Correct
C. 5
D. 7
Q15 — Fundamental Theorem of Arithmetic · easy · theory
The values of x and y in a factor tree where x branches into 3 and y, and y branches into 3 and 7, are:
A. x = 10, y = 14
B. x = 21, y = 63
C. x = 63, y = 21 ✓ Correct
D. x = 18, y = 9
Q16 — Fundamental Theorem of Arithmetic · hard · theory
If a = 2³ · 3, b = 2 · 3 · 5, c = 3ⁿ · 5 and LCM(a, b, c) = 2³ · 3² · 5, then n =:
A. 1
B. 2 ✓ Correct
C. 3
D. 4
Q17 — Fundamental Theorem of Arithmetic · medium · theory
If the prime factorization of a natural number n is 2³ · 3² · 5² · 7, how many consecutive zeros will n have at the end?
A. 2 ✓ Correct
B. 3
C. 4
D. 5
Q18 — Fundamental Theorem of Arithmetic · hard · theory
If n is a natural number, then 9²ⁿ - 4²ⁿ is always divisible by:
A. 5 only
B. 13 only
C. both 5 and 13 ✓ Correct
D. None of these
Q19 — Fundamental Theorem of Arithmetic · medium · theory
Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together?
A. 1 hour
B. 2 hours
C. 3 hours ✓ Correct
D. 4 hours
Q20 — Fundamental Theorem of Arithmetic · easy · theory
The ratio of LCM and HCF of the least composite and the least prime number is:
A. 1:2
B. 2:1 ✓ Correct
C. 1:1
D. 1:3
Q21 — Fundamental Theorem of Arithmetic · medium · theory
If HCF(306, 657) = 9, then LCM(306, 657) is:
A. 22338 ✓ Correct
B. 23328
C. 22833
D. 33228
Q22 — Fundamental Theorem of Arithmetic · medium · theory
The product of two consecutive natural numbers is always:
A. prime number
B. even number ✓ Correct
C. odd number
D. none of these
Q23 — Fundamental Theorem of Arithmetic · medium · theory
If a and b are co-prime numbers, then a² and b² are:
A. co-prime ✓ Correct
B. not co-prime
C. even
D. odd
Q24 — Fundamental Theorem of Arithmetic · medium · theory
If 2 is a factor of a², then 2 is also a factor of:
A. a ✓ Correct
B. 2a
C. a/2
D. a³
Q25 — Fundamental Theorem of Arithmetic · medium · theory
The HCF of two numbers is 23 and their LCM is 1449. If one number is 161, find the other:
A. 207 ✓ Correct
B. 307
C. 144
D. 230
Q26 — Fundamental Theorem of Arithmetic · easy · theory
Total number of factors of a prime number is:
A. 1
B. 2 ✓ Correct
C. 0
D. 3
Q27 — Fundamental Theorem of Arithmetic · easy · theory
The exponent of 2 in the prime factorization of 144 is:
A. 4 ✓ Correct
B. 5
C. 6
D. 3
Q28 — Fundamental Theorem of Arithmetic · hard · theory
The LCM of two numbers is 1200. Which of the following cannot be their HCF?
A. 600
B. 500 ✓ Correct
C. 400
D. 200
Q29 — Fundamental Theorem of Arithmetic · hard · theory
If n is any natural number, then 5²ⁿ - 1 is always divisible by:
A. 24 ✓ Correct
B. 23
C. 25
D. 26
Q30 — Fundamental Theorem of Arithmetic · easy · theory
The HCF of 3³ · 5 and 3² · 5² is:
A. 675
B. 45 ✓ Correct
C. 225
D. 15