Question

33333333333333333333×22222222222222222222222

Answer 1

33333333333333333333 = 3 × 11111111111111111111

22222222222222222222222 = 2 × 11111111111111111111111

33333333333333333333 × 22222222222222222222222

⇒ 3 × 11111111111111111111 × 2 × 11111111111111111111111

⇒ 6 × 11111111111111111111 × 11111111111111111111111

We have to find 11111111111111111111 × 11111111111111111111111

Some examples:

(1) = 11111 × 111 = 1233321

(2) = 1111111 × 11111 = 12345554321

(3) = 1111111111 × 1111 = 1234444444321

Let the first number at LHS be multiplicand and the other one be multiplier. Here the no. of digits of multiplicand should be greater than or equal to that of the multiplier. Let the no. of digits of multiplicand be m, and that of the multiplier be n where m ≥ n.

$\underbrace{11111...111}_m\ \times\ \underbrace{11111...111}_n\ =\ 1234......4321$

Some features are found here which are explained below:

  ⇒ The digit which is repeated many times in the middle of the product is the no. of digits of the multiplier.

$\underbrace{11111...111}_m\ \times\ \underbrace{11111...111}_n\ =\ 1234...nnnn...4321$

  ⇒ The no. of digit repeating itself at the middle of the product = No. of digit of multiplicand - No. of digit of multiplier + 1

$\underbrace{11111...111}_m\ \times\ \underbrace{11111...111}_n\ =\ 1234...\underbrace{nnnn}_{m-n+1}...4321$

  ⇒ The no. of digits of the product = No. of digits of the multiplicand + No. of digits of the multiplier - 1

$\underbrace{11111...111}_m\ \times\ \underbrace{11111...111}_n\ =\ \underbrace{1234...nnnn...4321}_{m+n-1}$

  ⇒ The no. of digits in the product from 1 at left to the repeating digit at rightmost = The no. of digits in the product from 1 at right to the repeating digit at leftmost = No. of digits of multiplicand

$\underbrace{11111...111}_m\ \times\ \underbrace{11111...111}_n\ =\ \underbrace{1234...nnnn}_{m}...4321\\ \\ \\ \underbrace{11111...111}_m\ \times\ \underbrace{11111...111}_n\ =\ 1234...\underbrace{nnnn...4321}_{m}$

Some products should be written decimally at first before writing its actual form. Decimally means to write a number by splitting the digits. Here the digits are split using a slash /.

E.g.: 1111111111111 × 111111111111111

Here the multiplier is the 13 digit number and the multiplicand, the 15 digit one.

Thus the product is written decimally as,

1/2/3/4/5/6/7/8/9/10/11/12/13/13/13/12/11/10/9/8/7/6/5/4/3/2/1

For removing the slashes and getting the real product, borrow the tens digit as remainders towards the nearest left digit, if the digit between the slashes is a two digit number. Repeat this process until there will only be 1 digit numbers between the slashes.

1/2/3/4/5/6/7/8/9/10/11/12/13/13/13/12/11/10/9/8/7/6/5/4/3/2/1

⇒ 1/2/3/4/5/6/7/8/9/₁0/₁1/₁2/₁3/₁3/₁3/₁2/₁1/₁0/9/8/7/6/5/4/3/2/1

⇒ 1/2/3/4/5/6/7/8/9+1/0+1/1+1/2+1/3+1/3+1/3+1/2+1/1+1/0/9/8/7/6/5/4/3/2/1

⇒ 1/2/3/4/5/6/7/8/10/1/2/3/4/4/4/3/2/0/9/8/7/6/5/4/3/2/1

⇒ 1/2/3/4/5/6/7/8/₁0/1/2/3/4/4/4/3/2/0/9/8/7/6/5/4/3/2/1

⇒ 1/2/3/4/5/6/7/8+1/0/1/2/3/4/4/4/3/2/0/9/8/7/6/5/4/3/2/1

⇒ 1/2/3/4/5/6/7/9/0/1/2/3/4/4/4/3/2/0/9/8/7/6/5/4/3/2/1

As all digits inside the slash become one digit each, remove the slash and thus the product is obtained.

⇒ 123456790123444320987654321

So,

11111111111111111111 × 11111111111111111111111

1/2/3/4/5/6/7/8/9/10/11/12/13/14/15/16/17/18/19/20/20/20/20/19/18/17/16/15/14/13/12/11/10/9/8/7/6/5/4/3/2/1

1/2/3/4/5/6/7/8/9+1/0+1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+2/0+2/0+2/0+2/0+1/9+1/8+1/7+1/6+1/5+1/4+1/3+1/2+1/1+1/0/9/8/7/6/5/4/3/2/1

1/2/3/4/5/6/7/8/10/1/2/3/4/5/6/7/8/9/11/2/2/2/1/10/9/8/7/6/5/4/3/2/0/9/8/7/6/5/4/3/2/1

1/2/3/4/5/6/7/8+1/0/1/2/3/4/5/6/7/8/9+1/1/2/2/2/1+1/0/9/8/7/6/5/4/3/2/0/9/8/7/6/5/4/3/2/1

1/2/3/4/5/6/7/9/0/1/2/3/4/5/6/7/8/10/1/2/2/2/2/0/9/8/7/6/5/4/3/2/0/9/8/7/6/5/4/3/2/1

1/2/3/4/5/6/7/9/0/1/2/3/4/5/6/7/8+1/0/1/2/2/2/2/0/9/8/7/6/5/4/3/2/0/9/8/7/6/5/4/3/2/1

1/2/3/4/5/6/7/9/0/1/2/3/4/5/6/7/9/0/1/2/2/2/2/0/9/8/7/6/5/4/3/2/0/9/8/7/6/5/4/3/2/1

Now, all are of 1 digit. But before removing the slashes, multiply 6 to get the answer for the question.

33333333333333333333 × 22222222222222222222222

6 × 11111111111111111111 × 11111111111111111111111

6(1/2/3/4/5/6/7/9/0/1/2/3/4/5/6/7/9/0/1/2/2/2/2/0/9/8/7/6/5/4/3/2/0/9/8/7/6/5/4/3/2/1)

6/12/18/24/30/36/42/54/0/6/12/18/24/30/36/42/54/0/6/12/12/12/12/0/54/48/42/36/30/24/18/12/0/54/48/42/36/30/24/18/12/6

6+1/2+1/8+2/4+3/0+3/6+4/2+5/4/0/6+1/2+1/8+2/4+3/0+3/6+4/2+5/4/0/6+1/2+1/2+1/2+1/2/0+5/4+4/8+4/2+3/6+3/0+2/4+1/8+1/2/0+5/4+4/8+4/2+3/6+3/0+2/4+1/8+1/2/6

7/3/10/7/3/10/7/4/0/7/3/10/7/3/10/7/4/0/7/3/3/3/2/5/8/12/5/9/2/5/9/2/5/8/12/5/9/2/5/9/2/6

7/3+1/0/7/3+1/0/7/4/0/7/3+1/0/7/3+1/0/7/4/0/7/3/3/3/2/5/8+1/2/5/9/2/5/9/2/5/8+1/2/5/9/2/5/9/2/6

7/4/0/7/4/0/7/4/0/7/4/0/7/4/0/7/4/0/7/3/3/3/2/5/9/2/5/9/2/5/9/2/5/9/2/5/9/2/5/9/2/6

So, all are 1 digit each. Now remove the slashes and we get

740740740740740740733325925925925925925926

Taking in international place value system,

740,740,740,740,740,740,733,325,925,925,925,925,925,926

This is the answer!