Find the greatest 5 digit number that is divisible by 67
The Correct Answer and Explanation is:
The greatest 5-digit number is 99999. To find the greatest 5-digit number divisible by 67, we divide 99999 by 67 and take the floor (i.e., the greatest whole number less than or equal to the result), then multiply it back by 67:
Step-by-step:
9999967≈1492.522\frac{99999}{67} \approx 1492.5226799999≈1492.522
Taking the floor:⌊1492.522⌋=1492\lfloor 1492.522 \rfloor = 1492⌊1492.522⌋=1492
Now multiply:1492×67=999641492 \times 67 = \boxed{99964}1492×67=99964
✅ Final Answer: 99964
📘 Explanation
To solve this problem, we need to find the greatest 5-digit number that is divisible by 67. A 5-digit number ranges from 10000 to 99999. So, the largest possible 5-digit number is 99999. We are looking for the highest number within this range that can be evenly divided by 67 — that is, one for which the remainder is zero.
The method begins with dividing 99999 by 67 to see how many times 67 can fit into it. This division gives a decimal number:99999÷67≈1492.52299999 \div 67 \approx 1492.52299999÷67≈1492.522
This result tells us that 67 can go into 99999 a little over 1492 times. Since we want the greatest whole number of times it goes in without exceeding 99999, we take the floor of this result, which is 1492.
Next, we multiply:1492×67=999641492 \times 67 = 999641492×67=99964
This result, 99964, is a 5-digit number, and importantly, it’s the largest such number divisible by 67.
If we tried 1493 × 67, we would get:1493×67=1000311493 \times 67 = 1000311493×67=100031
— which is a 6-digit number, and thus outside the range we want.
This method is efficient and reliable. By dividing the upper boundary (99999) and using the floor function, we directly pinpoint the highest multiple of 67 within the 5-digit range without checking each number manually.
