Find the greatest number of 5 digits that will give a remainder of 5 when divided by 8 and 9 respectively

The largest 5-digit number that leaves a remainder of 5 when divided by both 8 and 9 is 99,941. To find the greatest number of 5 digits that will give a remainder of 5 when divided by 8 and 9 respectively we need to use the concept of LCM (Least Common Multiple, which is 72 in this case) to find a general form for the numbers that satisfy the given conditions (x = 72k + 5). Then, we will find the largest 5-digit number that fits this form.

Complete Answer:

Here is a step by step process to find the greatest number of 5 digits that will give a remainder of 5 when divided by 8 and 9 respectively

Step 1: Understanding the condition

Let the required number be N .

• When N is divided by 8, it leaves a remainder of 5.
  Mathematically:

  N≡5 (mod 8)

  This means N−5 is divisible by 8.

• Similarly, when N is divided by 9, it leaves a remainder of 5.
  Mathematically:

  N≡5 (mod 9)

  This means N−5 is divisible by 9.

Hence, N−5 is a common multiple of 8 and 9.

 

Step 2: Finding the least common multiple (LCM) of 8 and 9

The LCM of 8 and 9 can be calculated as:

LCM(8,9)=72

Thus, N−5 is a multiple of 72.
Let N−5=72k, where k is a positive integer.
This gives:

N = 72 k + 5 N = 72k + 5  N=72k+5

 

Step 3: Finding the greatest 5-digit number

The greatest 5-digit number is 99999.
We want N=72k+5≤99999.
Subtract 5 from both sides:

72k≤99994

Now divide both sides by 72 to find k:

k≤7299994​≈1388.805

Since k must be an integer, take the greatest integer less than or equal to 1388.805:

k=1388

 

Step 4: Calculate $N$

Substitute k=1388 into N=72k+5:

N=72×1388+5=99936+5=99941

 

Step 5: Verify the solution

1. Check if 99941 leaves a remainder of 5 when divided by 8:

   99941÷8=12492 remainder 5

   This is correct.

2. Check if 99941 leaves a remainder of 5 when divided by 9:

   99941÷9=11104 remainder 5

   This is also correct.

Final Answer:

The greatest 5-digit number that leaves a remainder of 5 when divided by both 8 and 9 is: 99941​

Common Maths Questions:

What is 1 divided by 0?	How Many Thousands Make a Million?
Find the Mode of 10, 12, 11, 10, 15, 20, 19, 21, 11, 9, 10	What is the Divisibility Rule of 6?

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