Prime Factors HCF and LCM Word Problems

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Questions and Answers

Considering Lane and Rory's performance, formulate a diophantine equation that models the times at which they strike and pluck together, and then solve for the smallest positive integer solution, explaining your reasoning.

Let $9x = 15y$. The smallest solution occurs when $x=5$ and $y=3$. Therefore, they will next strike and pluck together after 45 seconds.

Propose an algorithm, using only Euclidean division and comparison, to determine the length of the ribbons David and Kylie can cut, proving that your algorithm always yields the greatest common divisor and relate this to Bezout's identity.

Use the Euclidean Algorithm to find the Greatest Common Divisor (GCD) of 180 and 225. $225 = 180 * 1 + 45$, $180 = 45 * 4 + 0$. Thus the GCD is 45. The longest length will be 45cm. Bezout's Identity states $ax + by = gcd(a,b)$

Given that $960 = 2^6 * 3 * 5$. Construct a formal proof, utilizing only the properties of prime factorization and divisibility, to demonstrate rigorously that 15 (3*5) is indeed the largest odd factor of 960.

Since $960 = 2^6 * 3 * 5$, any odd factor must be a product of only 3 and 5. The largest such is $3 * 5 = 15$. Any larger odd factor would require another prime, contradicting the prime factorization.

Assuming the amounts collected from Mr. Rooly and Mrs. Ottick represent the exact total revenue from the trip, derive a formal argument, based on the properties of divisibility, to prove that the highest possible cost per pupil corresponds to the greatest common divisor of £255 and £136. What is this cost, and how many pupils are in each class?

The GCF of 255 and 136 is 17. $255 = 17 * 15$ and $136 = 17 * 8$. The highest cost is £17. Mr. N Rooly has 15 pupils, Mrs. K Ottick has 8 pupils. Signup and view all the answers

Given HCF(a, b) = 32 and LCM(a, b) = 640, and knowing that $a * b = HCF(a, b) * LCM(a, b)$, determine all possible pairs (a, b) that satisfy these conditions. Further, derive a general formula for finding such pairs for any given HCF and LCM.

Since $ab = 32 * 640 = 20480$. We also know that $a = 32x$ and $b = 32y$ where x and y are coprime. Therefore, $32x * 32y = 20480$, so $xy = 20$. The coprime pairs of 20 are (1, 20) and (4, 5). Thus, the possible pairs (a, b) are (32, 640) and (128, 160). Signup and view all the answers

Formulate a system of congruences that models Cain and Chas washing their bedding on the same day, and use the Chinese Remainder Theorem to find the next date they will both wash their bedding if they both wash their bedding on April 1st. Provide a detailed explanation of each step in the application of the Chinese Remainder Theorem.

Cain washes bedding every 28 days (4 weeks) and Chas washes bedding every 5 days. We want to solve: $x ≡ 0 \pmod{28}$ and $x ≡ 0 \pmod{5}$. Since gcd(28, 5) = 1, we apply the CRT. The solution is $x ≡ 0 \pmod{140}$. Thus they wash together every 140 days. April 1st plus 140 days is August 19th. Signup and view all the answers

Using the definition of a square number and the uniqueness of prime factorization, construct a formal argument to demonstrate that if a number's prime factorization contains only even exponents, the number is a square number. Then, apply this principle to show that 5625 is a square number.

Since $5625 = 3^2 * 5^4 = (3 * 5^2)^2 = 75^2$, it is a square number. If a number's prime factorization contains only even exponents, it can be written as a square. Signup and view all the answers

Employ the properties of prime factorization and cube roots to derive a concise formula for determining the cube root of any number expressed as a product of prime factors with exponents divisible by 3. Apply this formula to show that $27000$ is a cube number, and explicitly calculate its cube root through prime factorization.

If $n = p_1^{3a} * p_2^{3b} * ...$, then the cube root of $n = p_1^a * p_2^b * ...$ $27000 = 2^3 * 3^3 * 5^3 = (2 * 3 * 5)^3 = 30^3$. Cube root is 30. Signup and view all the answers

Formulate a divisibility argument to determine all potential numbers of packets Sue-Lynn could have bought given that she has a total of 209 biscuits and each packet contains an equal number of biscuits. Justify why these are all the possibilities.

$209 = 11 * 19$. The factors of 209 are 1, 11, 19, 209. Sue-Lynn could have bought 1, 11, 19, or 209 packets. Signup and view all the answers

Develop an algorithm based on prime factorization for finding the smallest number divisible by all integers from 1 to 10 inclusive. Then, rigorously prove that the number generated by your algorithm is indeed the smallest possible number satisfying this divisibility condition.

Find the prime factorization of each number from 1 to 10. Then, take the highest power of each prime. $2^3 * 3^2 * 5 * 7 = 2520$. The smallest number is 2520. Signup and view all the answers

Flashcards

Lowest Common Multiple (LCM)

The smallest number that two or more numbers all divide into evenly.

Highest Common Factor (HCF)

The largest number that divides evenly into two or more numbers.