List the common multiples from 1 to 100 for each pair of numbers. Then find the least common multiple for each pair.
8 and 12, 3 and 15, 7 and 11, 9 and 10, 24 and 36, 20 and 25, 42 and 14, 30 and 12
Then find pairs of numbers for which the least common multiple equals the product of the two numbers.
Then look at the pairs of numbers you found in that part and tell what is true about all three pairs of numbers ?
Oh boy, let's dig into these multiples and see what we can find!
For 8 and 12:
Multiples: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96
Least common multiple: 24
For 3 and 15:
Multiples: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99
Least common multiple: 15
For 7 and 11:
Multiples: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98
Least common multiple: 77
For 9 and 10:
Multiples: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
Least common multiple: 90
For 24 and 36:
Multiples: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 456, 480, 504, 528, 552, 576, 600, 624, 648, 672, 696, 720, 744, 768, 792, 816, 840, 864, 888, 912, 936, 960, 984
Least common multiple: 72
For 20 and 25:
Multiples: 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, 680, 700, 720, 740, 760, 780, 800, 820, 840, 860, 880, 900, 920, 940, 960, 980
Least common multiple: 100
For 42 and 14:
Multiples: 42, 84, 126, 168, 210, 252, 294, 336, 378, 420, 462, 504, 546, 588, 630, 672, 714, 756, 798, 840, 882, 924, 966
Least common multiple: 42
For 30 and 12:
Multiples: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990
Least common multiple: 60
Now, let's find those magical pairs which have the least common multiple equal to the product of the two numbers:
8 and 12: Least common multiple = 24, Product = 96 (Not a match)
3 and 15: Least common multiple = 15, Product = 45 (Not a match)
7 and 11: Least common multiple = 77, Product = 77 (Bingo!)
9 and 10: Least common multiple = 90, Product = 90 (Bingo too!)
24 and 36: Least common multiple = 72, Product = 864 (No match)
20 and 25: Least common multiple = 100, Product = 500 (Nada)
42 and 14: Least common multiple = 42, Product = 588 (Nope)
30 and 12: Least common multiple = 60, Product = 360 (No match as well)
Alright, now what's true about those two pairs of numbers with matching least common multiples?
Well, we can see that for the pairs 7 and 11 and 9 and 10, the least common multiple equals the product of the two numbers. This is a super rare occurrence! Only these special pairs have this property.
But hey, don't feel bad if you can't find a life partner with whom your least common multiple equals your combined product. It's a tough ask for numbers, but easier for humans, I promise!
To find the common multiples for each pair of numbers, you can start by listing the multiples of each number and finding the numbers that appear in both lists.
For the pair 8 and 12:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
The common multiples for 8 and 12 are: 24, 48, 72, 96
Now, to find the least common multiple (LCM) for this pair, you can look for the smallest number that is divisible by both 8 and 12. In this case, the LCM is 24.
Repeat this process for the other pairs of numbers:
For 3 and 15:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120
The common multiples for 3 and 15 are: 15, 30, 45, 60, 75, 90, 105, 120
The LCM for 3 and 15 is 15.
Repeat this process for the other pairs of numbers and find their common multiples and LCMs:
For 7 and 11:
Common multiples: None (since 7 and 11 are prime numbers)
LCM: 7 x 11 = 77
For 9 and 10:
Common multiples: None
LCM: 9 x 10 = 90
For 24 and 36:
Common multiples: 72, 144
LCM: 72
For 20 and 25:
Common multiples: None
LCM: 20 x 25 = 500
For 42 and 14:
Common multiples: 42, 84, 126, 168
LCM: 42
For 30 and 12:
Common multiples: 60, 120
LCM: 60
To find pairs of numbers where the LCM equals the product of the two numbers, you can compare the LCMs you found to the products of the respective numbers. For example:
For 8 and 12:
LCM = 24
Product = 8 x 12 = 96
Since the LCM (24) is not equal to the product (96) for this pair, it does not satisfy the condition.
Repeat this step for the other pairs and see which ones satisfy the condition:
For 3 and 15:
LCM = 15
Product = 3 x 15 = 45
For 7 and 11:
LCM = 77
Product = 7 x 11 = 77
For 9 and 10:
LCM = 90
Product = 9 x 10 = 90
For the pairs that satisfy the condition, you can observe that in all three cases, the LCM is equivalent to the product of the two numbers.