What could be a general rule to multiply a number by 101 and write the product in one line? Extend this rule for multiplication by 1001, 10001, ...

Question

Homi · Accepted Answer

We can use the distributive property of multiplication to simplify these calculations.

Step 1 — Deriving the Rule for 101

Let $N$ be any number we want to multiply.

We want to calculate $N \times 101$ .

We know that 101 can be written as 100 + 1.

So, $N \times 101$ becomes $N \times (100 + 1)$ .

We use the distributive property. This property lets us multiply a number by a sum.

$N \times (100 + 1) = N \times 100 + N \times 1$

Multiplying $N$ by 100 means placing two zeros after $N$ .

For example, $3874 \times 100 = 387400$ .

Multiplying $N$ by 1 means the number $N$ stays the same.

For example, $3874 \times 1 = 3874$ .

So, the product $N \times 101$ is the sum of ( $N$ with two zeros) and ( $N$ ).

Let's use the example $3874 \times 101$ :

$3874 \times 101 = 3874 \times (100 + 1)$

$= 3874 \times 100 + 3874 \times 1$

$= 387400 + 3874$

$\boxed{391274}$

Diagram 1

Step 2 — General Rule for Multiplication by 101

From the previous step, we can form a general rule.

To multiply any number by 101:

First, write down the original number.

Next, write the same number again.

Shift this second copy two places to the right.

Add the two numbers together.

This method gives the product in one line.

Step 3 — Extending the Rule for 1001, 10001, and so on

Let us extend this idea for other numbers like 1001 or 10001.

For multiplication by 1001:

We can write 1001 as 1000 + 1.

So, $N \times 1001 = N \times (1000 + 1)$ .

Using the distributive property, this equals $N \times 1000 + N \times 1$ .

$N \times 1000$ means $N$ followed by three zeros.

So, we write the number $N$ .

Then, we write $N$ again, shifted three places to the right.

We add these two numbers.

For multiplication by 10001:

We can write 10001 as 10000 + 1.

So, $N \times 10001 = N \times (10000 + 1)$ .

Using the distributive property, this equals $N \times 10000 + N \times 1$ .

$N \times 10000$ means $N$ followed by four zeros.

So, we write the number $N$ .

Then, we write $N$ again, shifted four places to the right.

We add these two numbers.

We can observe a clear pattern here.

The number of zeros in the multiplier (like 1001 or 10001) tells us how many places to shift the second copy of the number.

For 101, there are two zeros, so we shift two places.

For 1001, there are three zeros, so we shift three places.

For 10001, there are four zeros, so we shift four places.

Answer

A general rule to multiply any number by 101 is:

• Write the number. • Write the same number again, but shifted two places to the right (like adding two zeros). • Add the two numbers. This gives the product in one line. Example: $3874 \times 101 = 387400 + 3874 = 391274$

This idea can be extended: For $\times 1001$ :

• Write the number. • Write the same number again, shifted three places to the right. • Add them.

For $\times 10001$ :

• Write the number. • Write the same number again, shifted four places to the right. • Add them.

So for multiplication by 101, 1001, 10001, … just write the number twice and the second copy is shifted according to the number of zeros.

Question 16

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