Questions and Answers
What is the prime factorization of the number represented by the product 38 × 2 × 25 × 39 × 50?
How many zeros are present in the product of 38 × 2 × 25 × 39 × 50?
What is the L.C.M. of the numbers 1296 and 2520?
Which of the following represents the H.C.F. of the numbers 1296 and 2520?
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Which is the minimum distance that three people walking 40 cm, 42 cm, and 45 cm per step must cover to be in sync?
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What is the largest number that can divide 1251, 9377, and 15628 while leaving remainders of 1, 2, and 3 respectively?
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When 16, 20, and 24 are divided by a certain smallest number, they leave a remainder of 5. What is that number?
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What is the expression representing the composite nature of the product 7 × 11 × 15 + 15?
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How do you define ascending order when comparing irrational numbers?
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What is the first step in comparing irrational numbers with different indexes?
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Which of the following best describes descending order for irrational numbers?
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What is required to convert irrational numbers to the same index?
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If you need to compare $3 ext{ }24$, $3 ext{ }56$, and $3 ext{ }29$, what is assumed about their radical values?
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What is the HCF of 26 and 91?
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What is the product of the two numbers 336 and 54?
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Which expression correctly represents the LCM of 26 and 91?
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What is the calculated HCF of 336 and 54?
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Verify the relationship between LCM and HCF for the numbers 510 and 92.
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Which collection of numbers is prime?
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What is the value of the LCM when calculating between 336 and 54?
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What relationship holds true for HCF and LCM of two numbers?
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What type of decimal expansion does the rational number $\frac{343}{73}$ have?
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What can be determined about the rational number $\frac{3125}{8}$?
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Which rational number will have a terminating decimal expansion?
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What type of decimal expansion does the rational number $\frac{129}{2 \times 5^7 \times 75}$ have?
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The rational number $\frac{17}{23}$ results in which type of decimal expansion?
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For the rational number $\frac{50}{210}$, what can be inferred about its decimal expansion?
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What is the decimal expansion type of the rational number $\frac{2^2 \times 5^2}{15}$?
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Which of the following options indicates a rational number with a non-terminating repeating decimal expansion?
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What is the HCF of 420, 130, and 600?
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Which of the following forms can a rational number have if its decimal expansion terminates?
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If the prime factorization of the denominator $b$ is not of the form $2^m5^n$, what type of decimal expansion does the rational number $ rac{a}{b}$ have?
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Which calculation is correct for finding HCF using a divisor?
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Which decimal number corresponds to the rational number $ rac{875}{10000}$?
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What can be deduced if a rational number has a prime factorization of the denominator $b$ as $2^m5^n$?
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Which statement correctly describes the transition from $ rac{a}{b}$ to its equivalent rational number using terminating decimals?
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Which example shows a rational number with a terminating decimal expansion?
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Study Notes
Prime Factorization and Zeros
- Number of zeros in the product of 38 × 2 × 25 × 39 × 50 is determined by the prime factors of 2 and 5.
- Prime factorization gives power of 2 as 3 and power of 5 as 4.
- Minimum(3, 4) = 3, resulting in three zeros.
LCM and HCF Calculation
- LCM of 1296 and 2520 can be derived from their prime factorizations.
- 1296 = (2^4 × 3^4) and 2520 = (2^3 × 3^2 × 5 × 7).
- LCM = (2^4 × 3^4 × 5 × 7 = 45360).
- HCF = (2^3 × 3^2 = 72).
HCF Application
- To find the largest number that divides 1251, 9377, and 15628 with respective remainders of 1, 2, and 3, adjust these values down to 1250, 9375, and 15625.
- HCF of these adjusted numbers is 625.
Composite Numbers
- 7 × 11 × 15 + 15 can be expressed in factored form to show it is composite: ((7 × 11 + 1) × 15).
- This approach validates the number as composite through identifiable factors.
HCF and LCM Product Verification
- To verify ( \text{HCF} × \text{LCM} = \text{Product of two numbers} ), calculations with pairs (e.g., 510 and 92) show this holds true.
- HCF(510, 92) = 10 and LCM(510, 92) = 23460 leads to confirmation of the identity.
Decimal Expansion Theorem
- For a rational number x = a/b, if the prime factorization of b is of the form (2^m5^n), then x has a terminating decimal expansion.
- Examples illustrate this with conversions demonstrating the factorization forms.
Irrational Numbers Comparison
- To organize irrational numbers in ascending or descending order, convert to a common index through the least common multiple.
- This method allows the comparison of differing- index radicals.
Termination of Decimal Expansions
- Rational numbers structured as ( \frac{p}{q} ) where q fits (2^m5^n) will have terminating decimal expansions.
- Numbers failing this test will yield non-terminating repeating decimals.
Rational Number Classification
- Distinguishing between terminating and non-terminating decimal expansions can be done through the structure of the denominator in fractions.
- This classification is fundamental for understanding and predicting decimal behavior in rational numbers.
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Description
This quiz covers key concepts in prime factorization, the calculation of least common multiples (LCM), and highest common factors (HCF). It includes practical applications of these concepts to solve problems. Enhance your understanding of composite numbers and their properties through engaging questions.
