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## SAT Number Properties Practice: Test of Divisibility

### Question 4: Numbers : Test of Divisibility & LCM

If K is divisible by 3, 4 and 5, which of the following will also divide K?

1. 3, 4, and 15
2. 12, 15, and 18
3. 5, 20, and 30
1. I only
2. II only
3. I and II only
4. I and III only
5. II and III only
Choice D. I and III only.

This number properties question tests your understanding of divisibility of numbers and LCM. Clues / hints to crack the question are given in the first tab. Using those try and solve the question yourself. If you are not able to solve using the hint, click on the next tabs to get a detailed explanation to this number properties question.

‘K’ is divisible by 3, 4, and 5. Try and answer the following questions to infer what kind of number K is. That should help you get to the solution to this question.

1. If a number K is divisible by another number d, will K be a multiple of d?
2. Is K the only such number or are there other numbers that will also be divisible by d?
3. If K is divisible by d, e, and f will it be a multiple of all three numbers?
4. What numbers, other than K, will also be divisible by all three numbers d, e, and f?

### Step 1: What is the least value of K that will be divisible by 3, 4, and 5?

If K is divisible by 3, 4, and 5, K has to be a multiple of 3, 4, and 5.

i.e., K is a common multiple of 3, 4, and 5.

The smallest or the first multiple common to 3, 4, and 5 is the LCM of 3, 4, and 5.

So, let us find the LCM of 3, 4, and 5.

Step 1: Prime factorize all the numbers.
3 is a prime number. 5 is also a prime number. When prime factorized, 4 will be written as 22.

Step 2: Write down all the prime factors in their respective highest power.
3, 22 and 5

Step 3: The LCM is the product of all different prime factors found in these numbers in their respective highest powers.
So, LCM of 3, 4 and 5 = 3 * 22 * 5 = 60.

So, K will be 60 or any multiple thereof.

Take a quick look at the choices to determine which of the three sets of numbers will also divide K to get the answer.

### Step 2: Check which of the choices divides 60

Option I: 3, 4, and 15
All these numbers will divide 60. So, Option I will be a part of the answer.

Option II: 12, 15, and 18
12 and 15 will divide 60. However, 18 will not. So, Option II is not correct.

Option III: 5, 20, and 30
All these numbers will divide 60. So, Option III will be a part of the answer.

Hence, the correct answer is Choice D – I and III only.