What is the Least Common Multiple or LCM of 34 and 96?

The least common multiple (LCM) of 34 and 96 is 1632.

Least common multiple or lowest common denominator (lcd) can be calculated in three simple and easy ways:

• LCM formula calculation method using greatest common factor (GCF)
• Prime factorization method
• List of multiples method

Let us look at each of these methods.

Calculating LCM of 34 and 96 using their GCF

The LCM of two numbers can be calculated using their GCF using the following formula: $LCM(a, b) = (a * b) / GCF(a, b)$

For this method, we need to first calculate the greatest common factor of 34 and 96, and then apply it to the formula equation above.

GCF of 34 and 96 is 2.

So, according to the formula,

$LCM(34, 96) = (34 x 96) / GCF(34, 96) = 3264 / 2 = 1632$.

Therefore, LCM(34, 96) = 1632.

Using Prime Factorization to calculate Least Common Multiple of 34 and 96

The least common multiple (LCM) of numbers can be also found by breaking up each number into its prime factors, and then multiplying the highest exponent prime factors of each number.

The steps are simple:

• Find prime factors of each number
• Denote each number as a multiple of prime factors in their exponential form
• Identify the highest exponential power for each prime factor for each number
• Multiply the prime factors with their highest powers to get the Least common multiple of the numbers

Let’s do this.

Prime factorization of 34

Prime factors of 34 are 2, 17.

So, prime factorization of 34 in exponential form would be:

$34 = 2^1 * 17^1$

Prime factorization of 96

Prime factors of 96 are 2, 3.

So, prime factorization of 96 in exponential form would be:

$96 = 2^5 * 3^1$

So, to calculate the LCM of 34 and 96, we would multiply the highest exponent prime factors:

$34 = 17^1 * 2^1 * 3^0$

$96 = 17^0 * 2^5 * 3^1$

So, LCM of 34 and 96 would be:

$LCM(34, 96) = 17^1 * 2^5 * 3^1 = 32 * 3 * 17 = 1632$

Thus, LCM of 34 and 96 is 1632.

LCM of 34 and 96 using list of multiples

This is the most rudimentary approach, but also the most tedious, especially when the numbers are large.

So, you should use this method only when the numbers are small. Otherwise, you should use the GCF method or the prime factorization method. This method is also not very efficient, so we would, in any case, recommend using the GCF method or the prime factorization method.

We will find and list the multiples of each number until we find a common multiple. This common multiple would therefore, by definition, be the lowest or least common multiple.

Multiples of 34:

34, 68, 102, 136, 170, 204, 238, 272, 306, 340, 374, 408, 442, 476, 510, 544, 578, 612, 646, 680, 714, 748, 782, 816, 850, 884, 918, 952, 986, 1020, 1054, 1088, 1122, 1156, 1190, 1224, 1258, 1292, 1326, 1360, 1394, 1428, 1462, 1496, 1530, 1564, 1598, 1632, 1666, 1700

Multiples of 96:

96, 192, 288, 384, 480, 576, 672, 768, 864, 960, 1056, 1152, 1248, 1344, 1440, 1536, 1632, 1728, 1824

As we can see, 1632 is the first number to be present in all lists of multiples.

Therefore, LCM(34, 96) = 1632