Finding the Smallest Number for Divisibility by 88

The question asks for the smallest number that needs to be added to 91673 to make the sum perfectly divisible by 88. To solve this, we need to find the remainder when 91673 is divided by 88.

Step 1: Understand Divisibility by 88

A number is divisible by 88 if it is divisible by both 8 and 11, as 8 and 11 are coprime factors of 88 ($8 \times 11 = 88$). We can find the remainder when 91673 is divided by 88 directly.

Step 2: Calculate the Remainder of 91673 divided by 88

Let's perform the division:

Divide 91673 by 88.

We can use long division or a calculator.

$91673 \div 88$

• $916 \div 88 = 10$ with a remainder of $916 - (88 \times 10) = 36$.
• Bring down the next digit (7) to make 367.
• $367 \div 88 = 4$ with a remainder of $367 - (88 \times 4) = 367 - 352 = 15$.
• Bring down the next digit (3) to make 153.
• $153 \div 88 = 1$ with a remainder of $153 - (88 \times 1) = 153 - 88 = 65$.

So, we can write the division as:

$91673 = 88 \times 1041 + 65$

The remainder when 91673 is divided by 88 is 65.

Step 3: Determine the Number to Add

We want to find the smallest number, let's call it '$x$', such that $91673 + x$ is exactly divisible by 88.

This means that $(91673 + x)$ should have a remainder of 0 when divided by 88.

We know that $91673$ has a remainder of 65 when divided by 88.

So, $(65 + x)$ must be the next multiple of 88, or 0.

To make the sum divisible by 88, we need to add a number that, when added to the current remainder (65), results in the next multiple of 88.

The next multiple of 88 after 65 is 88 itself.

Therefore, we need $65 + x = 88$.

Solving for $x$:

$x = 88 - 65$

$x = 23$

Step 4: Verification

Let's check if adding 23 to 91673 results in a number divisible by 88.

New sum = $91673 + 23 = 91696$

Now, divide 91696 by 88:

$91696 \div 88 = 1042$

Since the division results in a whole number (1042) with no remainder, the sum 91696 is perfectly divisible by 88.

The smallest number that should be added to 91673 is 23.