Question

You have piano lessons every 4th day and guitar lessons every 6th day. Today you have both lessons. In how many days will you have both lessons on the same day again? Explain?

Answer 1

12 because the Least Common Multiple of 6 and 4 is 12 6,12,18,24 4,8,12,16,20

Answer 2

$<p style="color:blue;"><u>This is application of least common multiples</u></p>$

Every 4th day means in multiples of 4 days, and every 6th day means in multiples of 6 days

If you had your piano lessons today, that means today is a day that is common to the 4th and 6th day.

Since the least common multiple of 4 and 6 is 12, you will have your next lesson on the 12th day.

You can use two approaches, to find the Least Common Multiples of 4 and 6.

$<b><u>Listing Method</u></b>$

Here, we list some few multiples of 4 and 6.

$[4,8,12,16,20,24,...] \\ \\ [6,12,18,24,30,...]$

Next, we list the common multiples,

$[12,24,...]$

Then we pick the least common multiple which is 12.

$<b><u>Prime Factorization Method</u></b>$

Here we express 4 and 6 as product of prime factors,

$4 = 2 \times 2 = {2}^{2} \\ \\ 6 = 2 \times 3$

Next, we multiply the common factors with the highest degrees.

$LCM = {2}^{2} \times 3 = 4 \times 3 = 12$