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

Ankita Singh

Updated on

07/02/2025

2 minute read

Answer

Verified

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:

$N$ $\equiv$ $5$ $($ $mod$ $8$ $)$

This means $N$ $-$ $5$ is divisible by 8.
Similarly, when $N$ is divided by 9, it leaves a remainder of 5.
Mathematically:

$N$ $\equiv$ $5$ $($ $mod$ $9$ $)$

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 $N$ $-$ $5$ $=$ $72$ $k$ , where $k$ is a positive integer.
This gives:

$N$ $=$ $72$ $k$ $+$ $5$

Step 3: Finding the greatest 5-digit number

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

$72$ $k$ $\leq$ $99994$

Now divide both sides by 72 to find $k$ :

$k$ $\leq$ $72$ $99994$ $$ $\approx$ $1388.805$

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 $N$ $=$ $72$ $k$ $+$ $5$ :

$N$ $=$ $72$ $\times$ $1388$ $+$ $5$ $=$ $99936$ $+$ $5$ $=$ $99941$

Step 5: Verify the solution

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

$99941$ $\div$ $8$ $=$ $12492$ $remainder$ $5$

This is correct.
Check if $99941$ leaves a remainder of 5 when divided by 9:

$99941$ $\div$ $9$ $=$ $11104$ $remainder$ $5$

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 $$

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