A 32 No worries! We‘ve got your back. Try BYJU‘S free classes today! B 36 No worries! We‘ve got your back. Try BYJU‘S free classes today! C 16 Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses D 20 No worries! We‘ve got your back. Try BYJU‘S free classes today!
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Solution The correct option is C 16 Note the following: Hence, 16 is the number with exactly 5 factors.
16=2×2×2×2, and hence has exactly 4+1=5 factors.
32=2×2×2×2×2, and hence has exactly 5+1=6 factors.
36=2×2×3×3, and hence has exactly (2+1)(2+1)=9 factors.
20=2×2×5, and hence has exactly (2+1)(1+1)=6 factors.
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Number of Factors of a Given Number
MATHEMATICS
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Understanding the number of factors a given number possesses involves delving into its prime factorization. I'll explain each concept mentioned in the article you provided:
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Factors of a Number: Factors of a number are the integers that can be multiplied together to give the original number. For instance, the factors of 16 are 1, 2, 4, 8, and 16.
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Prime Factorization: Breaking down a number into a product of its prime factors. For example, 16 = 2 × 2 × 2 × 2 and 36 = 2 × 2 × 3 × 3.
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Number of Factors: The number of factors a number has depends on its prime factorization. If a number is expressed as a product of prime factors, you can calculate the number of factors by adding 1 to the powers of each prime factor and then multiplying these incremented powers together.
For instance, 16 = 2^4, so the number of factors is (4 + 1) = 5. 32 = 2^5, thus having (5 + 1) = 6 factors. For 36 = 2^2 3^2, the number of factors is (2 + 1) (2 + 1) = 9.
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Smallest Divisible Number: Finding the smallest number divisible by a set of numbers involves finding their least common multiple (LCM). For example, the smallest five-digit number divisible by 12, 16, and 20 can be obtained by finding the LCM of these numbers.
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Number with a Specific Number of Factors: Determining the smallest number with a particular number of factors involves understanding factorization and prime factor powers. For example, finding the smallest number with exactly 8 factors involves considering the powers of primes in its factorization.
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Numbers Divisible by Prime Numbers: Finding the smallest number divisible by the first five prime numbers (2, 3, 5, 7, 11) requires calculating their LCM.
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Numbers with Specific Divisibility Properties: For instance, finding the smallest number that, when decreased by 7, is exactly divisible by a set of numbers (12, 16, 18, 21, and 28) involves understanding their relationship and working backward to find the required number.
These concepts are integral to understanding the factors of a number, finding the smallest numbers meeting specific criteria, and understanding divisibility properties. They're foundational in various mathematical problems and play a crucial role in number theory and problem-solving in mathematics.
