CS 064 Primes and Hash Functions

Which memory locations are assigned by the hashing function h(k) = k mod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 104578690 is

Compute k mod 101 by dividing by 101 and finding the remainders. This can be done with a calculator that keeps 13 digits of accuracy internally. Just divide the number by 101, subtract the integer part of the answer, and multiply the fraction that remains by 101. The result will be almost exactly an integer, and that integer is the answer. For 104578690, the answer is 58.

Which memory locations are assigned by the hashing function h(k) = k mod 101 to the records of insurance company customers with the following Social Security numbers? The assigned memory location to 432222187 is

Compute k mod 101 by dividing by 101 and finding the remainders. This can be done with a calculator that keeps 13 digits of accuracy internally. Just divide the number by 101, subtract the integer part of the answer, and multiply the fraction that remains by 101. The result will be almost exactly an integer, and that integer is the answer. For 432222187, the answer is 60.

Identify the prime factorization of 10!

The definition of factorial gives us the factorization 10! = 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2. We now replace each composite factor by its prime factorization. 10! = (2^1 · 5^1) · 3^2 · 2^3 · 7^1 · (2^1 · 3^1) · 5^1 · 2^2 · 3^1 · 2^1 and add the exponents of each unique prime to find the prime factorization 10! = 2^8 · 3^4 · 5^2 · 7.

Identify the greatest common divisor of the following pair of integers. 2^6 · 3^2 · 5 · 7 · 11^1 · 13 and 2^11 · 3^6 · 11^2 · 17

The greatest common divisor is calculated finding the minimum exponent for each prime factor. In the given integers, 2^6, 3^2, and 11^1 are the minimum exponents for each prime factor. So, the greatest common divisor is 2^6 · 3^2 · 11^1.

Identify the greatest common divisor of the following pair of integers. 2^3 · 3^4 · 5^5 and 2^1 · 3^2 · 5^2

The greatest common divisor is found by finding the minimum exponent for each prime factor. In the given integer 2^1, 3^2, and 5^2 are the minimum exponent for each prime factor. So, the greatest common divisor is 2^1 · 3^2 · 5^2.

What is the relation of the integers 14, 25, 85?

The integers 14, 25, and 85 are not pairwise relatively prime because gcd(25, 85) = 5.

What is the relation of the integers 21, 34, 53?

The integers 21, 34, and 53 are pairwise relatively prime because 21, 34, and 53 are prime. Hence, no pair of these numbers have a common prime factor.

Find the least common multiple of each of these pair of integers. 2^5 · 3^1 · 5^2 and 2^3 · 3^2 · 5^3

The least common multiple is formed by finding the maximum exponent for each prime factor. In the given integers 2^5, 3^2, and 5^3 provide the maximum exponent for each prime factor. So, the least common multiple is 2^5 · 3^2 · 5^3.

Find the least common multiple of each of these pair of integers. 20, 20^20

The least common multiple is formed by finding the maximum exponent for each prime factor. In the given integers, 20^20 provide the maximum exponent for each prime factor. So, the least common multiple is 20^20.

Find the least common multiple of each of these pair of integers. 2^2 · 3^3 · 5 · 7 · 11^2 · 13 and 2^9 · 3^8 · 11^1 · 17^14

The least common multiple is formed by finding the maximum exponent for each prime factor. In the given integers, 2^9, 3^8, 5, 7, 11^2, 13, and 17^14 provide the maximum exponents for each prime factor. So, the least common multiple is 2^9 · 3^8 · 5 · 7 · 11^2 · 13 · 17^14.

Identify the positive integers that are not relatively prime to 18

The unique prime factors of 18 are 2 and 3. Therefore, all integers that are divisible by 2 or by 3 are not relatively prime to 18.

How to Identify if an integer is prime.

To determine if each integer is prime, divide it by successive primes beginning with 2 and ending with the square root of the number.

The greatest prime factor of 103 is

To find the prime factorization of each integer, first perform divisions of the integers by successive primes, beginning with 2. 103 is a prime number.

The greatest prime factor of 15 is

To find the prime factorization of each integer, first perform divisions of the integers by successive primes, beginning with 2. 15 = 3 · 5

The greatest prime factor of 16 is

To find the prime factorization of each integer, first perform divisions of the integers by successive primes, beginning with 2. 16 = 2·2·2·2 = 24.

The smallest prime factor of 85 is

To find the prime factorization of each integer, first perform divisions of the integers by successive primes, beginning with 2. 85 = 5 · 17.