Math Quiz: Decimals and Factors
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Questions and Answers

What is the nature of the number calculated by the expression $5 × 11 × 13 + 7$?

  • None
  • Composite number
  • Prime number
  • Odd number (correct)

Which expression always yields a number that ends with the digit 6?

  • 2n
  • 4n
  • 6n (correct)
  • 8n

For two positive rational numbers, $a$ and $b$ (where $a ≠ b$), what type of number is $(√a + √b)(√a - √b)$?

  • Irrational number (correct)
  • $(√a - √b)^2$
  • Zero
  • Rational number

If $p$ is a positive rational number which is not a perfect square, then what type of number is $-3p$?

<p>Rational number (B)</p> Signup and view all the answers

Which of the following is true in the Euclidean Division Lemma when $a = bq + r$?

<p>$0 ≤ r ≤ b$ (D)</p> Signup and view all the answers

What is always true about the Highest Common Factor (HCF) in relation to the Least Common Multiple (LCM)?

<p>Factor of L.C.M. (B)</p> Signup and view all the answers

The product of two consecutive natural numbers is always which of the following?

<p>An even number (C)</p> Signup and view all the answers

For the expression $pn = (a × 5)^n$ to end with the digit zero, what should $a$ be for natural number $n$?

<p>Even number (B)</p> Signup and view all the answers

Which of the following is a property of the Highest Common Factor (HCF)?

<p>HCF is a factor of the LCM (C)</p> Signup and view all the answers

What can be said about the product of two consecutive natural numbers?

<p>It is always an even number (C)</p> Signup and view all the answers

For which of the following expressions would the output always be an irrational number?

<p>$√2 + √3$ (C)</p> Signup and view all the answers

If $p$ is a positive rational number that is not a perfect square, what can we conclude about $-3p$?

<p>It is always a rational number (D)</p> Signup and view all the answers

Which condition must hold true in Euclid's Division Lemma for $a = bq + r$?

<p>$0 ≤ r &lt; b$ (C)</p> Signup and view all the answers

What type of number is always represented by $√a - √b$ when $a$ and $b$ are two positive rational numbers and $a ≠ b$?

<p>It is always an irrational number (C)</p> Signup and view all the answers

Determine the nature of the decimal representation of $3.131131113...$.

<p>It is a rational number (A)</p> Signup and view all the answers

Which expression is guaranteed to yield a number that is always positive?

<p>$√a + √b$ (A)</p> Signup and view all the answers

Flashcards

Irrational number between 0 and 1?

An irrational number is a number that cannot be expressed as a fraction of two integers. Examples include √2, π, and some repeating decimals that don't terminate.

HCF and LCM relationship

The Highest Common Factor (HCF) of two numbers is always a factor of their Lowest Common Multiple (LCM).

Product of two consecutive natural numbers?

The product of two consecutive natural numbers is always even.

What always ends with 6?

Numbers ending with the digit 6 will always be related to the digit 6, which typically means multiples of 6 or potentially 4 multiplied by a number, or multiples of 6.

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(√a + √b)(√a - √b)

For positive rational numbers a and b (where a ≠ b), (√a + √b)(√a - √b) results in a rational number.

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Identifying prime vs composite number?

A prime number is a whole number greater than 1 that has only two factors – 1 and itself. A composite number is a whole number greater than 1 that has more than two factors.

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Euclid Division Lemma condition?

In the Euclid division lemma, 'a = bq + r', where 'a' and 'b' are positive integers, '0 ≤ r < b'.

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pn ending in zero condition?

For pn = (a × 5)n to end with zero, 'a' must be an even natural number.

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What is a prime number?

A prime number is a whole number greater than 1 that is only divisible by 1 and itself. Examples include 2, 3, 5, 7, 11, and so on.

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Composite number

A composite number is a whole number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, 10, and so on.

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What does Euclid's Division Lemma state?

Euclid's Division Lemma states that for any two positive integers 'a' and 'b', there exist unique integers 'q' (quotient) and 'r' (remainder) such that 'a = bq + r', where '0 ≤ r < b'.

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How do you know if a number is irrational?

An irrational number cannot be expressed as a fraction of two integers. They have infinite decimal representations that don't repeat.

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HCF and LCM relation

The Highest Common Factor (HCF) is always a factor of the Lowest Common Multiple (LCM) of two numbers.

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Product of consecutive natural numbers

The product of two consecutive natural numbers is always an even number. At least one of the consecutive numbers will be even, and any number multiplied by an even number results in an even number.

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What makes a number end in 6?

A number ends in 6 if it is a multiple of 6. For example, 6, 12, 18, 24, and so on.

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Study Notes

Multiple Choice Questions - Study Notes

  • Terminating Decimal Expansion: Certain fractions have decimal expansions that end. Numbers like 37/45, 21/25 , and 17/49 do not have terminating decimal expansions. 22/32 and 89/49 are examples of non-terminating decimal expansions.

  • HCF and LCM: The highest common factor (HCF) and lowest common multiple (LCM) of numbers are calculated to find relationships between them. For example: the HCF x LCM for 50 and 20 is 1000.

  • Euclid's Division Lemma: This principle states that given two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' so that a=bq+r where 0 ≤ r less than b. The remainder must be 0 or a positive number smaller than 'b'.

  • Rational Number Decimal Expansion: Rational numbers have decimal expansions which either terminate (e.g., 22.5) like numbers in Q5, or repeat in a pattern. Fractions with denominators having only 2 and/or 5 as prime factors will terminate in their decimal representation, like 22.5 /32 or 11/22 .

  • Co-prime Numbers: Two numbers are co-prime (or relatively prime) if their greatest common divisor (GCD) or highest common factor (HCF) is 1. If p and q are co-prime, then HCF(p, q)=1.

  • Prime Numbers and LCM: If p and q are prime numbers, then the LCM(p, q) is pq.

  • HCF and LCM relationship: HCF(a, b) × LCM(a, b)=a ×b, for any two positive integers a, and b. For example, HCF(2520, 6600) = 40 and LCM(2520, 6600) = 252k. If one knows these values and the relationship for a and b , one can find k by multiplying the given values in this way.

  • Non-Terminating, Repeating Decimals: Not all rational numbers have terminating decimal expansions. If a fraction's denominator includes prime factors other than 2 and/or 5, its decimal expansion will repeat. For example, 35/14 which is 2.5 has a terminating decimal expansion. On the other hand, 7/12 has a non-terminating and repeating decimal.

  • Prime Factorization: Finding the prime factors of a number is important in calculating their HCF and LCM. For example, 5005 = 5 x 7 x 11 x 13 has 4 prime factors.

  • Odd/Even Integers: Integers can be expressed in specific forms using parameters, for example, every even integer is 2m. And an odd integer can be expressed as 2q+1.

  • Coprime Numbers and Squares: If two positive integers a and b are coprime, then a² and b² are also coprime (i.e., HCF(a², b²)=1).

  • Least Prime Factor: The least prime factor of a number is the smallest prime number that divides that number evenly.

  • Irrational Numbers: Irrational numbers have decimals that do not repeat and do not terminate. For example, √2, π and √5 are irrational numbers.

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Test your knowledge on decimal expansions, HCF, LCM, and Euclid's Division Lemma in this math quiz. Understand the characteristics of rational numbers and how to calculate their decimal representations. Perfect for students looking to strengthen their foundation in mathematics.

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