a. Look at the data in Tables factors of a given number 1 and 2. How is the number of rectangular arrays? b. Can there be any other numbers in column A? Why? c. The numbers that go in column B are called prime numbers. If a number is in column B, what are its factors? d. The numbers that go in column C are composite numbers. Describe the factors of a composite number.
Added by Jonathan S.
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Problem 1. Find the prime factorization of the following numbers: (a) 269. (b) 311. (c) 721. (d) 1001. (e) 20!. (f) gcd(20!, 2^8 × 3^5 × 7^10 × 334). (g) lcm(20!, 2^8 × 3^5 × 7^10 × 334). (h) gcd(p^a × q^b × r^c, p^d × q^e × r^f), where p, q, r are distinct positive primes and a, b, c, d, e, f are non-negative integers. (i) lcm(p^a × q^b × r^c, p^d × q^e × r^f), where p, q, r are distinct positive primes and a, b, c, d, e, f are non-negative integers. Problem 2. Find the total number of positive divisors of the following numbers: (a) 2^3 × 3^5. (b) 2^10 × 3^20. (c) p^a × q^b, where p, q are distinct positive primes and a, b are non-negative integers. (d) p1^a1 × · · · × pn^an, where p1, . . . , pn are distinct positive primes and a1, . . . , an are nonnegative integers. (e) 669. Problem 3. Find the sum of all positive divisors of 669. Problem 5. Show that a^2 | b^2 if and only if a|b. Hint: use prime factorization for the hard implication.
Sri K.
Please explain what it means to describe a number as a "prime number" and a "composite number". 2. Give an example of a prime and a composite number and explain why such numbers are prime/composite. Please provide a detailed explanation and numerical examples for each answer.
Julie S.
Use this definition: A prime number is a positive whole number with no factors other than itself and $1 .$ For example, $2,13,$ and 37 are primes, but 24 and 39 are not. $B y$ convention 1 is not considered prime, so the list of the first few primes is as follows: $2,3,5,7,11,13,17,19,23,29, \ldots$ Let $f$ be the function that assigns to each natural number $x$ the number of primes that are less than or equal to $x .$ For example, $f(12)=5$ because, as you can easily check, five primes are less than or equal to $12 .$ Similarly, $f(3)=2$ because two primes are less than or equal to $3 .$ Find $f(8)$ $f(10),$ and $f(50)$
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