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ISEE Upper Level Quantitative : Least Common Multiple

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #1 : How To Find The Least Common Multiple

What is the least common multiple of 15 and 25?

Possible Answers:

150

175

125

Correct answer:

Explanation:

$\textrm{LCM }(15,25)$ 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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Example Question #1 : How To Find The Least Common Multiple

N= M + 1

Which is the greater quantity?

(a) LCM(M, 100)

(b) LCM(N, 100)

Possible Answers:

Not enough information is given to answer the question.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Correct answer:

Not enough information is given to answer the question.

Explanation:

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

M = 2, N = 3 and M = 3, N = 4

and divide into 100 , so LCM(2, 100) = 100 and LCM(4, 100) = 100 .

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

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

Therefore, if M = 2, N = 3 , (b) is greater, and if M = 3, N = 4 , (a) is greater.

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Example Question #1 : How To Find The Least Common Multiple

Which is the greater quantity?

(a) $50 \times 60$

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

Possible Answers:

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

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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Example Question #2 : How To Find The Least Common Multiple

Which is the greater quantity?

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

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

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

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

(b) To find GCF (6,8,10) , list their factors:

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

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

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

GCF (6,8,10) = 2

To find LCM (6,8,10) ,examine their prime factorizations:

6 = 2 $\times 3$

$8 = 2 \times 2 \times 2$

10: 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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Example Question #2 : Least Common Multiple

Which of the following is the greater quantity?

(A) The least common multiple of 25 and 30

(B) 300

Possible Answers:

(B) is greater

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(A) is greater

Correct answer:

(B) is greater

Explanation:

To find LCM(25,30) we can list some multiples of both numbers and discover the least number in both lists:

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

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

LCM(25,30) = 150 < 300 , so (B) is greater

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Example Question #2 : Least Common Multiple

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

Possible Answers:

160

250

150

200

Correct answer:

200

Explanation:

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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Example Question #3 : How To Find The Least Common Multiple

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

Possible Answers:

52,416

157,248

104,832

314,496

628,992

Correct answer:

314,496

Explanation:

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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Example Question #7 : How To Find The Least Common Multiple

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

Possible Answers:

abcde

$a^{2}bd$

$a^{2}b^{2}c^{2}d^{2}e^{2}$

abd

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

Correct answer:

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

Explanation:

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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Example Question #8 : How To Find The Least Common Multiple

Define an operation $\bigstar$ as follows:

For all positive integers and

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

Evaluate $100 \bigstar 80$ .

Possible Answers:

516

1,008

180

240

420

Correct answer:

420

Explanation:

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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Example Question #6 : How To Find The Least Common Multiple

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

Possible Answers:

160

250

150

200

Correct answer:

200

Explanation:

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 200 .

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Least Common Multiple

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Find The Least Common Multiple

Example Question #1 : How To Find The Least Common Multiple

Example Question #1 : How To Find The Least Common Multiple

Example Question #2 : How To Find The Least Common Multiple

Example Question #2 : Least Common Multiple

Example Question #2 : Least Common Multiple

Example Question #3 : How To Find The Least Common Multiple

Example Question #7 : How To Find The Least Common Multiple

Example Question #8 : How To Find The Least Common Multiple

Example Question #6 : How To Find The Least Common Multiple

All ISEE Upper Level Quantitative Resources

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