Running laps

Number Theory Level 1

Alvin, Calvin and Melvin are running laps around a track. Alvin runs a lap every $4$ minutes, Calvin runs a lap every $6$ minutes and Melvin runs a lap every $9$ minutes. If they start off at the same point, how many minutes will it take before all three of them are back at the starting point?

Details and assumptions:
- A lap of the track refers to 1 complete round, back to the starting point.

36 72 90 54

2 solutions

Arron Kau Staff
May 13, 2014

Since Alvin returns to the starting point every $4$ minutes, Calvin returns every $6$ minutes and Melvin returns every $9$ minutes, we seek the lowest common multiple of $4$ , $6$ and $9$ . This is equal to $2^2 \times 3^2 = 36$ .

Jack Rawlin
Dec 27, 2014

This can be solved by finding $lcm (4, 6, 9)$ .

This is actually pretty easy. Just use the ladder method.

$\div$	a	b	c
-	4	6	9

First divide by any prime which goes into at least one of the above numbers, let's start with $2$

$\div$	a	b	c
-	4	6	9
2	2	3	9

Since $9$ doesn't divide by $2$ we carry it down. Next let's divide by $3$

$\div$	a	b	c
-	4	6	9
2	2	3	9
3	2	1	3

Now we only have primes left (and a $1$ ) so we multiple them together.

$2\cdot3\cdot2\cdot3 = 36$

Running laps

2 solutions

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