ISEE Upper Level Quantitative Reasoning - 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 $100 \bigstar 80$ .

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:

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

Which of the following is the least common multiple of and

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 .

Which of the following is the least common multiple of 6, 9, and 12?

648

Explanation

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.

Find the LCM of and

Explanation

LCM is the least common multiple of the pair. There are a couple of terms at play here, so first find the LCM of 12 and 48. Since 12 goes into 48, 48 is the LCM. Then, look at xy and x. XY is the LCM between those two. So, multiply those together to get as your answer.

Which is the greater quantity?

(a) $50 \times 60$

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

(a) and (b) are equal.

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

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.

What is the least common multiple of 15 and 25?

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:

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

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

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

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.

Which of the following is the greater quantity?

(A) The least common multiple of 25 and 30

(B) 300

(B) is greater

(A) is greater

(A) and (B) are equal

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

Explanation

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

Which is the greater quantity?

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

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

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation

(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.

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

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

$a^{2}bd$

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

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: