What number should be subtracted from 65 to make it divisible by 6, 12, and 15
The Correct Answer and Explanation is:
To find the number that should be subtracted from 65 to make it divisible by 6, 12, and 15, we follow a systematic approach.
✅ Correct Answer: 5
📘 Step-by-Step Explanation
To solve the problem, we need to subtract a number from 65 so that the result is divisible by 6, 12, and 15. This means the resulting number must be a common multiple of 6, 12, and 15.
The first step is to find the Least Common Multiple (LCM) of the three numbers:
🔹 Prime Factorizations:
- 6 = 2 × 3
- 12 = 2² × 3
- 15 = 3 × 5
To find the LCM, we take the highest powers of all prime numbers involved:
- 2² (from 12)
- 3 (common in all)
- 5 (from 15)
LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60
So, the smallest number that is divisible by 6, 12, and 15 is 60.
Now, we ask:
What number should be subtracted from 65 to get 60?
That’s simply:
65 – 60 = 5
So, 5 should be subtracted from 65.
✅ Verification
Let’s check that 60 is divisible by 6, 12, and 15:
- 60 ÷ 6 = 10
- 60 ÷ 12 = 5
- 60 ÷ 15 = 4
All results are whole numbers, confirming 60 is divisible by all three.
🧠 Final Thought
This type of problem tests your understanding of the Least Common Multiple and basic arithmetic reasoning. Instead of checking all numbers below 65, using the LCM method saves time and ensures accuracy. This approach is especially useful when dealing with multiple divisibility conditions.
