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QuestionFind the least common multiple (LCM) for these pairs: a) 12, 14 b) 21, 45 c) 45, 60 d) 38, 42 e) 32, 45 f) 28, 52.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the least common multiple (LCM) of each pair of numbers. . The LCM of two integers a and b is the smallest positive integer that is divisible by both a and b.

STEP 2

We will use the formula for the least common multiple, which is given byLCM(a,b)=∣a⋅b∣gcd(a,b)LCM(a, b) = \frac{|a \cdot b|}{gcd(a, b)}where gcd(a, b) is the greatest common divisor of a and b.

STEP 3

First, we will find the LCM of12 and14. The gcd of12 and14 is2, so we can use the formulaLCM(12,14)=∣12⋅14∣2LCM(12,14) = \frac{|12 \cdot14|}{2}

STEP 4

Calculate the LCM of12 and14.
LCM(12,14)=∣12⋅14∣2=84LCM(12,14) = \frac{|12 \cdot14|}{2} =84

STEP 5

Next, we will find the LCM of21 and45. The gcd of21 and45 is3, so we can use the formulaLCM(21,45)=∣21⋅45∣3LCM(21,45) = \frac{|21 \cdot45|}{3}

STEP 6

Calculate the LCM of21 and45.
LCM(21,45)=∣21⋅45∣3=315LCM(21,45) = \frac{|21 \cdot45|}{3} =315

STEP 7

Next, we will find the LCM of45 and60. The gcd of45 and60 is15, so we can use the formulaLCM(45,60)=∣45⋅60∣15LCM(45,60) = \frac{|45 \cdot60|}{15}

STEP 8

Calculate the LCM of45 and60.
LCM(45,60)=∣45⋅60∣15=180LCM(45,60) = \frac{|45 \cdot60|}{15} =180

STEP 9

Next, we will find the LCM of38 and42. The gcd of38 and42 is2, so we can use the formulaLCM(38,42)=∣38⋅42∣2LCM(38,42) = \frac{|38 \cdot42|}{2}

STEP 10

Calculate the LCM of38 and42.
LCM(38,42)=∣38⋅42∣2=798LCM(38,42) = \frac{|38 \cdot42|}{2} =798

STEP 11

Next, we will find the LCM of32 and45. The gcd of32 and45 is, so we can use the formulaLCM(32,45)=∣32⋅45∣LCM(32,45) = \frac{|32 \cdot45|}{}

STEP 12

Calculate the LCM of32 and45.
LCM(32,45)=∣32⋅45∣=1440LCM(32,45) = \frac{|32 \cdot45|}{} =1440

STEP 13

Finally, we will find the LCM of28 and52. The gcd of28 and52 is, so we can use the formulaLCM(28,52)=∣28⋅52∣LCM(28,52) = \frac{|28 \cdot52|}{}

STEP 14

Calculate the LCM of28 and52.
LCM(28,52)=∣28⋅52∣4=364LCM(28,52) = \frac{|28 \cdot52|}{4} =364So, the least common multiples of the pairs area)84b)315c)180d)798e)1440f)364

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