Counting in fours
The multiples of 4 are the numbers you land on when you count in fours: 4, 8, 12, 16, 20, 24, 28, and on without end. Each one is 4 times a whole number: 12 = 4 × 3 and 28 = 4 × 7.
0 jumps of 4 = 0
Count on to 24 in jumps of 4
Count on in jumps of 4 and every landing is ringed: 4, 8, 12, 16, 20 and 24.
Counting in sixes
The multiples of 6 come from counting in sixes: 6, 12, 18, 24, 30, and on. Each one is 6 times a whole number: 18 = 6 × 3.
0 jumps of 6 = 0
Count on to 24 in jumps of 6
In jumps of 6 the landings are 6, 12, 18 and 24. Two of them, 12 and 24, were landings in the count of fours as well.
Where both counts land
Compare the two counts. 12 is in both, and so is 24: 12 = 4 × 3 = 6 × 2, and 24 = 4 × 6 = 6 × 4. A number that is a multiple of both of two numbers is a common multiple of them. Up to 30, the common multiples of 4 and 6 are 12 and 24.
The counts go on meeting after 30: at 36, 48, 60, and on, every 12. The common multiples of 4 and 6 are the multiples of 12.
Up to 30, only 12 and 24 are on the count of fours and on the count of sixes.
The first meeting
The smallest common multiple is the first number both counts land on. For 4 and 6 it is 12. It is called the Lowest Common Multiple, or LCM for short. The LCM of 4 and 6 is 12.
Multiplying the two numbers always gives a common multiple, because 4 × 6 is 4 counted six times and also 6 counted four times. It need not be the first one: 4 × 6 = 24, and the counts have already met at 12.
Three hops of 4 reach 12, and 12 is also two sixes. It comes before 24, so it is the lowest common multiple.
On both counts, not one
A common multiple must be on both counts, so test each candidate against both numbers. 16 is a multiple of 4, but 16 ÷ 6 = 2 remainder 4, so it is not a multiple of 6. 18 is a multiple of 6, but 18 ÷ 4 = 4 remainder 2.
Multiples and factors go in opposite directions. The multiples of 6 start at 6 and grow: 6, 12, 18. The factors of 6, which are 1, 2, 3 and 6, are never more than 6.
Stepping on by the LCM
The counts of 5 and 7 first meet at 35, so the common multiples of 5 and 7 are 35, 70, 105, and on, each 35 more than the one before.
Adding a common multiple does not change what is left over. 18 in groups of 5 is 3 groups with 3 left over, and in groups of 7 it is 2 groups with 4 left over. 35 is a whole number of fives and a whole number of sevens, so 18 + 35 = 53 has the same 3 left over in groups of 5 and the same 4 left over in groups of 7.
Worked example: LCM with Shifting Non-Uniform Remainders
Question A florist has a collection of roses numbering fewer than 100. When she arranges them in bunches of 5, she has 3 roses left over. When she arranges them in bunches of 7, she has 4 roses left over. What are all the possible total numbers of roses she could have?
1.Column 1 (Groups of 7 plus 4): 4, 11, 18, 25, 32, 39...
Bunches of 7 with 4 over: 4, 11, 18, 25, 32, ... 2.Column 2 (Groups of 5 plus 3): 3, 8, 13, 18, 23, 28...
Bunches of 5 with 3 over: 3, 8, 13, 18, 23, ... 3.Identify the first intersection in both lists: 18.
The lists first agree at 18. 4.Calculate repeating block: 5 × 7 = 35.
Both patterns repeat together every 5 × 7 = 35. 5.Step forward by 35: 18, (18 + 35 = 53), (53 + 35 = 88).
So the answers are 18, 53 and 88. 6.Stop since the next number 88 + 35 = 123 is greater than 100.
88 + 35 = 123 is more than 100, so those three are all.
Answer: 18, 53 or 88 roses
Common mistakes
- Listing only the first value (18) and missing the subsequent values (53, 88) requested by 'all possible numbers'.
- Adding the remainders together (3 + 4 = 7) and trying to apply a constant offset approach.
- Stepping forward by the sum of divisors (5 + 7 = 12) instead of their LCM (35).