Multiple surprises
Solution by :
Yuvaan Sonika, Grade 5D, Inventure Academy, Bangalore, India
Here are some challenges involving consecutive numbers and multiples.
Can you find three consecutive numbers where the first is a multiple of 2, the second is a multiple of 3 and the third is a multiple of 4?
Yes, 2,3 and 4 are the easiest consecutive numbers to find that are multiples of 2,3,4. So, the numbers themselves are the first set of consecutive numbers.
Can you find several examples?
To find other examples, I thought of all multiples of 4 between 0-40 and identified the set of consecutive numbers like 14-15-16, 26-27-28 and 38-39-40 which met the criteria of the first being multiple of 2, second being multiple of 3 and third being multiple of 4. All the other multiples of 4 (8,12,20,24,32,36) were not followed by a multiple of 2 and 3.
What do you notice?
Can you explain your findings?
I noticed a pattern that each set of numbers was increasing by 12 from the previous set. For example- 26-27-28 are 12 more than 14-15-16 respectively and so is 14-15-16 from 2-3-4.
I found from this pattern that 12 is in fact the lowest common multiple (LCM) of 2,3 and 4.
So, we can find all possible consecutive numbers that are multiples of 2,3 and 4 if we keep on adding the LCM to 2,3 and 4. Also, the numbers themselves form the first set of consecutive numbers.
What if the first is a multiple of 3, the second is a multiple of 4, and the third is a multiple of 5?
Using the above found method, we can find all the consecutive sets of multiples of 3,4 and 5 by adding the LCM which is 60 in this case to these numbers. So, the possible consecutive set of numbers are the numbers 3-4-5 themselves and after adding LCM the other possible sets are 63-64-65, 123-124-125, 183-184-185 and so on.
What if the first is a multiple of 4, the second is a multiple of 5, and the third is a multiple of 6?
Again, we can find all the consecutive sets of multiples of 4,5 and 6 by adding the LCM which is 60 again in this case to these numbers. So, the consecutive set of numbers are 4-5-6, 64-65-66, 124-125-126, 184-185-186 and so on.
Is there a way to find sets of four consecutive numbers which are multiples of 2, 3, 4, and 5 (in this order)?
Yes, same method can be used to find sets of four consecutive numbers which are multiples of 2,3,4 and 5 by adding the LCM to these numbers. LCM of these four numbers is 60 and the possible sets of consecutive numbers that are multiples of 2,3,4 and 5 are 2-3-4-5, 62-63-64-65, 122-123-124-125, 182-183-184-185 and so on.
Or five consecutive numbers which are multiples of 2, 3, 4, 5, and 6 (in this order)?
Yes, same method can be used to find sets of five consecutive numbers which are multiples of 2,3,4,5 and 6 by adding the LCM to these numbers. LCM of these five numbers is also 60 and the possible sets of consecutive numbers that are multiples of 2,3,4,5 and 6 are 2-3-4-5-6, 62-63-64-65-66, 122-123-124-125-126, 182-183-184-185-186 and so on.
Can you use what you have discovered to help you find a few sets of ten consecutive numbers in which:
- the first is a multiple of 1
- the second is a multiple of 2
- the third is a multiple of 3
- the fourth is a multiple of 4
- the fifth is a multiple of 5
- the sixth is a multiple of 6
- the seventh is a multiple of 7
- the eighth is a multiple of 8
- the ninth is a multiple of 9
- the tenth is a multiple of 10?
Yes, the same method can be used to find sets of ten consecutive numbers which are multiples of 1,2,3,4,5,6,7,8,9 and 10 by adding the LCM to these numbers. LCM of these ten numbers is 2520 and the possible sets of consecutive numbers that are multiples of 1,2,3,4,5,6,7,8,9 and 10 are:
1-2-3-4-5-6-7-8-9-10,
2521-2522-2523-2524-2525-2526-2527-2528-2529-2530 ,
5041-5042-5043-5044-5045-5046-5047-5048-5049-5050 and so on.