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

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Question

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

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

What is the least common multiple of 15 and 25?

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

Which is the greater quantity?

(a)

(b)

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

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

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

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.

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 $a^{2}bde^{2}$ .

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 $100 \bigstar 80$ .

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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Question

Which of the following is the least common multiple of and

Answer

List the first few multiples of both and $40\textup:$

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

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

The least number in both lists of factors is .

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Question

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

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?

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 6, 9, and 12?

Answer

The least common multiple is the smallest number in value that is a multiple of all three of the numbers. The best way to find the least common multiple (LCM) is to make a quick list of the first few multiples of each number, and then identify the smallest number that is common to all three lists.

Multiples are numbers that you get when multiplying the original number by other numbers.

18 is an answer choice that is common to both 6 and 9, but it is not also a multiple of 12, so it is not correct.

The smallest value that is a common multiple of all three is 36, so this is the LCM.

While 648 is a multiple of all three numbers, it is not the least common multiple of the three numbers.

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Question

$\frac{3}{5} + \frac{4}{7} - \frac{1}{3}=$

Answer

This equation should be solved from left to right, finding the least common denominator for each pair. First we find the least common denominator of and , which is . The fractions then become and . is then subtracted from that sum:

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Question

What is the least common multiple of and ?

Answer

Find all of the factors of both 9 and 6:

$9=3\times3$

$6=2\times3$

Eliminate all factors that 9 and 6 have in common, in this instance 3 (only eliminate the repeated instance). Multiply all factors that are not in common:

$LCM=3\times2\times3=18$

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Question

Which of the following values of makes this statement true?

Answer

For to be a true statement, must be a multiple of . Each of the four values given is divisible by , as seen below:

$160 \div 20 = 8$

$320 \div 20= 16$

$440 \div 20 = 22$

$720 \div 20= 36$

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Question

Which of the following values of makes this statement false?

Answer

For to be true, must be a factor of . Of the five choices, all are factors of except for . This is the correct choice.

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Question

Which of the following values of makes this statement true?

Answer

For to be a true statement, must be a multiple of 15.

$250 \div 15 = 16 \textrm{ R }10$

$375 \div 15= 25$

$425 \div 15 = 28 \textrm{ R }5$

$655 \div 15 = 43 \textrm{ R }10$

$950 \div 15 = 63 \textrm{ R } 5$

Only 375 yields no remainder when divided by 15.

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Question

What is the least common multiple of 15 and 18?

Answer

To find the least common multiple, you need to determine the multiple that both numbers share that is of the least value. List the multiples of each number and identify the first number (least value) that is in both lists:

The LCM of 15 and 18 is 90 since it is the first number that shows up in both lists.

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Question

What is the least common multiple of 4, 12, and 16?

Answer

To find the least common multiple, you need to determine the multiple that all three numbers share that is of the least value. List the multiples of each number and identify the first number (least value) that is in both lists:

The LCM of 4, 12, and 16 is 48 since it is the first number that shows up in all three lists.

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