How to find the least common multiple - ISEE Upper Level Quantitative Reasoning

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Question

What is the least common multiple of 15 and 25?

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Answer

$\textrm{LCM }(15,25)$ is the lowest number that is a multiple of both 15 and 25, so we see which is the first number that appears in both lists of multiples.

The multiples of 15:

$\left \{ 15, 30, 45, 60, \underline{75}, 90, 105,...\right \}$

The multiples of 25:

$\left \{ 25, 50, \underline{75}, 100, 125, 150,...\right \}$

$\textrm{LCM }(15,25)= 75$

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Question

30 is the least common mutiple of 10 and which of these numbers?

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Answer

9, 12, and 20 can be eliminated immediately, since none of these have 30 as a multiple \[30 divided by any of them yields a remainder\].

since 10 is a multiple of 5, so 5 can be eliminated.

The set of multiples of 10 is $\left \{ 10,20,\underline{30},40,50...\right \}$ and the set of multiples of 15 is $\left \{ 15, \underline{30}, 45, 60,...\right \}$ ; since 30 is the smallest number that appears on both lists, .

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Question

What is the least common multiple of 16 and 20?

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Answer

$\textrm{LCM }(16,20)$ is the lowest number that is a multiple of both 16 and 20, so we see which is the first number that appears in both lists of multiples.

The multiples of 16:

$\left \{ 16, 32, 48, 64, \underline{80}, 96, 112,...\right \}$

The multiples of 20:

$\left \{ 20, 40, 60, \underline{80}, 100, 120,...\right \}$

$\textrm{LCM }(16,20) = 80$

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Question

What is the least common multiple of 15 and 25?

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Answer

$\textrm{LCM }(15,25)$ is the lowest number that is a multiple of both 15 and 25, so we see which is the first number that appears in both lists of multiples.

The multiples of 15:

$\left \{ 15, 30, 45, 60, \underline{75}, 90, 105,...\right \}$

The multiples of 25:

$\left \{ 25, 50, \underline{75}, 100, 125, 150,...\right \}$

$\textrm{LCM }(15,25)= 75$

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Question

30 is the least common mutiple of 10 and which of these numbers?

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Answer

9, 12, and 20 can be eliminated immediately, since none of these have 30 as a multiple \[30 divided by any of them yields a remainder\].

since 10 is a multiple of 5, so 5 can be eliminated.

The set of multiples of 10 is $\left \{ 10,20,\underline{30},40,50...\right \}$ and the set of multiples of 15 is $\left \{ 15, \underline{30}, 45, 60,...\right \}$ ; since 30 is the smallest number that appears on both lists, .

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Question

What is the least common multiple of 16 and 20?

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Answer

$\textrm{LCM }(16,20)$ is the lowest number that is a multiple of both 16 and 20, so we see which is the first number that appears in both lists of multiples.

The multiples of 16:

$\left \{ 16, 32, 48, 64, \underline{80}, 96, 112,...\right \}$

The multiples of 20:

$\left \{ 20, 40, 60, \underline{80}, 100, 120,...\right \}$

$\textrm{LCM }(16,20) = 80$

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Question

What is the least common multiple of 15 and 25?

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Answer

$\textrm{LCM }(15,25)$ is the lowest number that is a multiple of both 15 and 25, so we see which is the first number that appears in both lists of multiples.

The multiples of 15:

$\left \{ 15, 30, 45, 60, \underline{75}, 90, 105,...\right \}$

The multiples of 25:

$\left \{ 25, 50, \underline{75}, 100, 125, 150,...\right \}$

$\textrm{LCM }(15,25)= 75$

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Question

30 is the least common mutiple of 10 and which of these numbers?

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Answer

9, 12, and 20 can be eliminated immediately, since none of these have 30 as a multiple \[30 divided by any of them yields a remainder\].

since 10 is a multiple of 5, so 5 can be eliminated.

The set of multiples of 10 is $\left \{ 10,20,\underline{30},40,50...\right \}$ and the set of multiples of 15 is $\left \{ 15, \underline{30}, 45, 60,...\right \}$ ; since 30 is the smallest number that appears on both lists, .

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Question

What is the least common multiple of 16 and 20?

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Answer

$\textrm{LCM }(16,20)$ is the lowest number that is a multiple of both 16 and 20, so we see which is the first number that appears in both lists of multiples.

The multiples of 16:

$\left \{ 16, 32, 48, 64, \underline{80}, 96, 112,...\right \}$

The multiples of 20:

$\left \{ 20, 40, 60, \underline{80}, 100, 120,...\right \}$

$\textrm{LCM }(16,20) = 80$

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Question

What is the least common multiple of 15 and 25?

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Answer

$\textrm{LCM }(15,25)$ is the lowest number that is a multiple of both 15 and 25, so we see which is the first number that appears in both lists of multiples.

The multiples of 15:

$\left \{ 15, 30, 45, 60, \underline{75}, 90, 105,...\right \}$

The multiples of 25:

$\left \{ 25, 50, \underline{75}, 100, 125, 150,...\right \}$

$\textrm{LCM }(15,25)= 75$

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Question

30 is the least common mutiple of 10 and which of these numbers?

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Answer

9, 12, and 20 can be eliminated immediately, since none of these have 30 as a multiple \[30 divided by any of them yields a remainder\].

since 10 is a multiple of 5, so 5 can be eliminated.

The set of multiples of 10 is $\left \{ 10,20,\underline{30},40,50...\right \}$ and the set of multiples of 15 is $\left \{ 15, \underline{30}, 45, 60,...\right \}$ ; since 30 is the smallest number that appears on both lists, .

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Question

What is the least common multiple of 16 and 20?

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Answer

$\textrm{LCM }(16,20)$ is the lowest number that is a multiple of both 16 and 20, so we see which is the first number that appears in both lists of multiples.

The multiples of 16:

$\left \{ 16, 32, 48, 64, \underline{80}, 96, 112,...\right \}$

The multiples of 20:

$\left \{ 20, 40, 60, \underline{80}, 100, 120,...\right \}$

$\textrm{LCM }(16,20) = 80$

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Question

Which of the following is the least common multiple of 25 and 40?

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Answer

List the first few multiples of both 25 and 40:

$25: 25, 50, 75, 100, 125, 150, 175, \underline{200},...$

$40: 40, 80, 120, 160, \underline{200}, 240,...$

The least number in both lists of factors is 200.

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Question

Which is the greater quantity?

(a)

(b)

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Answer

We show that the given information is not enough by taking two cases:

and

and divide into , so and .

is prime and $100 = 2\times2\times5\times5$ , so

$LCM(3, 100) = 2\times2\times3\times5\times5 = 300$ .

Therefore, if , (b) is greater, and if , (a) is greater.

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Question

Which is the greater quantity?

(a) $50 \times 60$

(b) $LCM (50,60 ) \times GCF (50,60 )$

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Answer

The prime factorizations of 50 and 60 are:

$50 = 2 \times 5 \times 5$

$60 = 2\times2\times3\times5$

The greatest common factor of 50 and 60 is the product of the prime factors they share:

$GCF (50,60) = 2 \times 5 = 10$

The least common multiple of 50 and 60 is the product of all of the prime factors, with shared factors counted once:

$LCM (50,60 ) = 2\times2\times3\times5\times 5 = 300$

$GCF (50,60) \times LCM (50,60) = 10 \times 300 = 3,000 = 50 \times 60$ ,

(a) and (b) are equal.

Note: it is also a property of the integers that the product of the GCF and the LCM of two integers is equal to the product of the two integers themselves.

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Question

Which is the greater quantity?

(a) $6 \times 8 \times 10$

(b) $GCF (6,8,10) \times LCM (6,8,10)$

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Answer

(a) $6 \times 8 \times 10 = 480$

(b) To find , list their factors:

$6: 1, \underline{2}, 3, 6$

$8: 1,\underline{2}, 4, 8$

$10:1,\underline{2},5,10$

To find ,examine their prime factorizations:

$\times 3$

$8 = 2 \times 2 \times 2$

$\times 5$

$LCM (6,8,10) = 2 \times 2 \times 2 \times 3 \times 5 = 120$

$GCF (6,8,10) \times LCM (6,8,10) = 2 \times 120 = 240$

(a) is greater.

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Question

Which of the following is the greater quantity?

(A) The least common multiple of 25 and 30

(B) 300

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Answer

To find we can list some multiples of both numbers and discover the least number in both lists:

$25: 25, 50, 75, 100, 125, \underline{150},...$

$30:30, 60, 90, 120, \underline{150}, 180...$

, so (B) is greater

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Question

Multiply the least common multiple of 504 and 624 by the greatest common factor of 504 and 624.

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Answer

The product of the least common multiple of any two integers and the greatest common factor of the same two integers is the product of the two integers themselves. Therefore,

$LCD (504, 624) \times GCF (504, 624) = 504 \times 624 = 314,496$ .

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Question

, , , , and are five distinct prime integers. Give the least common multiple of $abc^{2}d$ and $a^{2}bde^{2}$ .

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Answer

If two integers are broken down into their prime factorizations, their greatest common factor is the product of the prime factors that appear in one or both factorizations.

Since , , , , and are distinct prime integers, the two expressions can be factored into their prime factorizations as follows - with their common prime factors underlined:

$abc^{2}d ; ; = a; ; ; ; \cdot b \cdot c \cdot c \cdot d$

$a^{2}bde^{2} = a \cdot a \cdot b ; ; ; ;; ; ; ; \cdot d \cdot e \cdot e$

The LCM collects each of the factors:

$LCM = a \cdot a \cdot b \cdot c \cdot c \cdot d \cdot e \cdot e$

$= a^{2}bc^{2}de^{2}$

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Question

Define an operation $\bigstar$ as follows:

For all positive integers and

$a \bigstar b = \textup{lcd }(a,b) + \textup{gcf }(a,b)$ .

Evaluate $100 \bigstar 80$ .

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Answer

To find the LCD and GCF of 100 and 80, first, find their prime factorizations:

$80 =\underline{2} \cdot\underline{ 2} \cdot 2 \cdot 2 \cdot \underline{5}$

$100 = \underline{2} \cdot \underline{2} \cdot \underline{5} \cdot 5$

The GCF of the two is the product of their shared prime factors, so

$\textup{gcf }(100, 80) = 2 \cdot 2 \cdot 5 = 20$

The LCM is the product of all factors that occur in one or the other factorization, so

$\textup{lcm }(100, 80) = 2 \cdot 2\cdot 2\cdot 2 \cdot 5 \cdot 5 = 400$

Add:

$a \bigstar b = \textup{lcd }(a,b) + \textup{gcf }(a,b) = 400 + 20 = 420$

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