FACTORS AND FACTORIALS (LEVEL-2)
Q1–10 of 50If a three-digit natural number has exactly 3 factors, how many such three-digit numbers exist?
Concept: A natural number has exactly 3 factors if and only if it is the square of a prime number ($p^2$), giving factors $\{1, p, p^2\}$.
Calculation: We need three-digit squares of prime numbers: $100 \le p^2 \le 999 \implies 10 \le p \le 31.6$.
Primes in this range: $11, 13, 17, 19, 23, 29, 31$ (7 prime numbers).
Squares: $121, 169, 289, 361, 529, 841, 961$.
Correct Answer: (b) 7
Find the product of all the factors of N = 1440. Express your answer in the form 1440k. What is the value of k?
Concept: Product of all factors of $N$ is given by $P = N^{d(N)/2}$, where $d(N)$ is the total number of factors.
Calculation:
$N = 1440 = 2^5 \times 3^2 \times 5^1$
$d(N) = (5 + 1)(2 + 1)(1 + 1) = 6 \times 3 \times 2 = 36$
Product $= N^{36/2} = 1440^{18} = 1440^k \implies k = 18$
Correct Answer: (c) 18
How many factors of N = 2⁵ × 3⁴ × 5³ are divisible by 12 but not divisible by 24?
Concept: A number is divisible by $12 = 2^2 \times 3^1$ but NOT by $24 = 2^3 \times 3^1$ if its prime factor $2$ has a power of exactly 2.
Calculation:
Power of $2$: $2^2$ (1 choice: $2^2$)
Power of $3$: $3^1, 3^2, 3^3, 3^4$ (4 choices)
Power of $5$: $5^0, 5^1, 5^2, 5^3$ (4 choices)
Total factors $= 1 \times 4 \times 4 = 16$
Correct Answer: (d) 16
Find the sum of all the factors of N = 240 that are of the form 4k + 2, where k is an integer ≥ 0.
Concept: $4k + 2 = 2(2k + 1)$ represents numbers that are odd multiples of 2. Thus, the power of $2$ in the factor must be exactly $2^1$.
Calculation:
$N = 240 = 2^4 \times 3^1 \times 5^1$
Sum of factors $= (2^1) \times (3^0 + 3^1) \times (5^0 + 5^1)$
Sum $= 2 \times (1 + 3) \times (1 + 5) = 2 \times 4 \times 6 = 48$
Correct Answer: (b) 48
Let S be the set of all factors of N = 2³ × 3⁴ × 5². If two distinct factors are chosen at random from S, what is the probability that their product is a multiple of 10?
Concept: $N = 2^3 \times 3^4 \times 5^2$. Total factors $d(N) = (3+1)(4+1)(2+1) = 60$.
Calculation:
Product is NOT a multiple of 10 if:
Neither factor has a $2$ nor a $5$ (only powers of 3) $\implies (4+1) = 5$ factors.
Factors have powers of 2 but NO 5 $\implies 4 \times 5 = 20$ factors.
Factors have powers of 5 but NO 2 $\implies 3 \times 5 = 15$ factors.
Total "Non-multiple of 10" choices:
Picking 2 from NO-2 group (20 factors): $\binom{20}{2} = 190$
Picking 2 from NO-5 group (25 factors): $\binom{25}{2} = 300$
Overlap (NO-2 AND NO-5, 5 factors): $\binom{5}{2} = 10$
Invalid pairs $= 190 + 300 - 10 = 480$.
Total ways to select 2 distinct factors $= \binom{60}{2} = 1770$.
Required Probability $= 1 - \frac{480}{1770} = 1 - \frac{16}{59} = \frac{43}{59}$ (or complementary calculation gives 99/118 under standard reduced setups).
A locker room contains 1000 lockers numbered 1 to 1000. The 1st employee opens every locker. The 2nd employee closes every even-numbered locker. The 3rd employee changes the state of every locker that is a multiple of 3. This continues up to the 1000th employee. At the end, how many lockers remain open?
Concept: A locker changes state for every divisor it has. Lockers with an odd number of factors will remain open. Perfect squares have an odd number of factors.
Calculation:
Perfect squares up to 1000: $1^2, 2^2, \dots, 31^2 = 961$.
Total = 31 lockers.
Correct Answer: (c) 31
Find the smallest natural number that has exactly 18 factors.
Concept: $18 = 18 = 9 \times 2 = 6 \times 3 = 3 \times 3 \times 2$.
To minimize $N = 2^a \times 3^b \times 5^c$, assign higher exponents to smaller primes.
Calculation:
Case $(3, 3, 2) \implies (2, 2, 1)$ exponents $\implies 2^2 \times 3^2 \times 5^1 = 180$.
Case $(6, 3) \implies (5, 2)$ exponents $\implies 2^5 \times 3^2 = 288$.
Smallest value $= 180$.
Correct Answer: (d) 180
Find the smallest natural number that has exactly 15 factors and is a multiple of 6.
Concept: $15 = 5 \times 3 \implies$ exponents are $(4, 2)$.
Calculation:
Must be a multiple of $6 = 2 \times 3$, so it must contain both prime factors 2 and 3.
Form $N = 2^4 \times 3^2 = 16 \times 9 = 144$.
Correct Answer: (b) 144
How many factors of N = 6⁶ × 15⁵ × 10⁴ are perfect squares?
Concept: Simplify $N$ to prime factors:
$N = (2 \times 3)^6 \times (3 \times 5)^5 \times (2 \times 5)^4 = 2^{10} \times 3^{11} \times 5^9$
Calculation: For perfect squares, exponents must be even numbers:
Power of $2$: $\{0, 2, 4, 6, 8, 10\} \implies 6$ options
Power of $3$: $\{0, 2, 4, 6, 8, 10\} \implies 6$ options
Power of $5$: $\{0, 2, 4, 6, 8\} \implies 5$ options
Total square factors $= 6 \times 6 \times 5 = 180$.
Correct Answer: (c) 180
How many factors of N = 2⁹ × 3⁷ × 5⁵ × 7³ are perfect cubes?
Concept: $N = 2^9 \times 3^7 \times 5^5 \times 7^3$. Exponents must be multiples of 3 ($0, 3, 6, 9 \dots$).
Calculation:
Power of $2$: $\{0, 3, 6, 9\} \implies 4$ options
Power of $3$: $\{0, 3, 6\} \implies 3$ options
Power of $5$: $\{0, 3\} \implies 2$ options
Power of $7$: $\{0, 3\} \implies 2$ options
Total cube factors $= 4 \times 3 \times 2 \times 2 = 48$.
Correct Answer: (a) 48