What is the largest five digit number that is exactly divisible by 95?

Introduction / Context:
This problem asks for the largest five digit number that is a multiple of 95. It combines understanding of place value for five digit numbers with divisibility and simple multiplication. The aim is to find the highest multiple of 95 that does not exceed 99999.

Given Data / Assumptions:
The largest five digit number is 99999.We want the greatest multiple of 95 that is less than or equal to 99999.All numbers involved are integers.

Concept / Approach:
If we divide 99999 by 95 and ignore any remainder, the integer part of the quotient tells us how many full multiples of 95 fit into 99999. Multiplying that integer by 95 gives the required largest multiple that is still within the five digit range.

Step-by-Step Solution:
Step 1: Compute the approximate quotient of 99999 divided by 95.Step 2: Start with 95 × 1000 = 95000, which is well below 99999.Step 3: The difference 99999 − 95000 = 4999 shows how much more we can add.Step 4: Compute 95 × 50 = 4750 and 95 × 52 = 4940.Step 5: Add 52 to 1000 to get 1052. Then 95 × 1052 = 95000 + 4940 = 99940.Step 6: If we try 95 × 1053, we get 99940 + 95 = 100035, which exceeds 99999 and is no longer a five digit number.Step 7: Thus, 99940 is the largest five digit multiple of 95.

Verification / Alternative check:
Check the divisibility of 99940 by 95. Divide 99940 by 5 first: 99940 / 5 = 19988. Next, check divisibility of 19988 by 19. Multiply 19 by 1000 to get 19000, and 19 by 526 to get 9994, so 19 × 1052 = 19988. Therefore 99940 = 95 × 1052 exactly. Since 95 × 1053 overshoots 99999, 99940 is indeed the largest valid multiple.

Why Other Options Are Wrong:
The number 99935 is not divisible by 95; dividing by 5 gives 19987, which is not a multiple of 19. The numbers 99936 and 99933 are not multiples of 5 at all because they do not end in 0 or 5. Therefore they cannot be multiples of 95, which is 5 × 19. Only 99940 satisfies both the divisibility and the maximum size conditions.

Common Pitfalls:
Some learners attempt random trial and error close to 99999 without using division, which may miss the exact multiple. Others forget that a multiple of 95 must end in 0 or 5 due to the factor 5. Using the stepwise method of first multiplying 95 by a convenient base like 1000 and then adjusting upward is efficient and reliable.

Final Answer:
The largest five digit number divisible by 95 is 99940.