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Multiplication by 111, 1111, ... (Speed Math)

Prerequisite: multiplication by $11$

Example 1: Calculate 752 × 111

Add digits of $752$ like this (add up to the depth of $3$ as the number of ones in $111$ is $3$)

$7\quad 7+5\quad 7+5+2\quad 5+2\quad 2$

Rewrite with the sum of digits

$7\quad 12\quad \class{bd p-5}{1}4\quad 7\quad 2$

In $14$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $12$

$7\quad \class{bd p-5}{1}3\quad 4\quad 7\quad 2$

In $13$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $7$

$8\quad 3\quad 4\quad 7\quad 2$

answer $=83472$

Example 2: Calculate 57 × 111

Since $111$ has $3$ ones, write $57$ as $057.$ Add digits of $057$ like this (add up to the depth of $3$ as the number of ones in $111$ is $3$)

$0\quad 0+5\quad 0+5+7\quad 5+7\quad 7$

Rewrite with the sum of digits

$0\quad 5\quad 12\quad \class{bd p-5}{1}2\quad 7$

In $12$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $12$

$0\quad 5\quad \class{bd p-5}{1}3\quad 2\quad 7$

In $13$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $5$

$0\quad 6\quad 3\quad 2\quad 7$

answer $=06327=6327$

Example 3: Calculate 2573 × 111

Add digits of $2573$ like this (add up to the depth of $3$ as the number of ones in $111$ is $3$)

$2\quad 2+5\quad 2+5+7\quad 5+7+3\quad 7+3\quad 3$

Rewrite with the sum of digits

$2\quad 7\quad 14\quad 15\quad \class{bd p-5}{1}0\quad 3$

In $10$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $15$

$2\quad 7\quad 14\quad \class{bd p-5}{1}6\quad 0\quad 3$

In $16$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $14$

$2\quad 7\quad \class{bd p-5}{1}5\quad 6\quad 0\quad 3$

In $15$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $7$

$2\quad 8\quad 5\quad 6\quad 0\quad 3$

answer $=285603$

Example 4: Calculate 1257 × 1111

Add digits of $1257$ like this (add up to the depth of $4$ as the number of ones in $1111$ is $4$)

$1\quad 1+2\quad 1+2+5\quad 1+2+5+7\quad 2+5+7\quad 5+7\quad 7$

Rewrite with the sum of digits

$1\quad 3\quad 8\quad 15\quad 14\quad \class{bd p-5}{1}2\quad 7$

In $12$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $14$

$1\quad 3\quad 8\quad 15\quad \class{bd p-5}{1}5\quad 2\quad 7$

In $15$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $15$

$1\quad 3\quad 8\quad \class{bd p-5}{1}6\quad 5\quad 2\quad 7$

In $16$ shown above, take $1$ as carry. Add this $1$ to its left neighbor number $8$

$1\quad 3\quad 9\quad 6\quad 5\quad 2\quad 7$

answer $=1396527$

Comments(19)

profileHimanshu
2015-09-11 03:09:34 
i didn't get ans. of this
938*111 pls solve
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profileRaj
2015-09-11 11:32:38 
938*111

9  (9+3)  (9+3+8)  (3+8)  8

9  (2, c=1)  (0, c=2)  (1, c=1)  8

10  4  1  1  8

i.e., 104118
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profiletejaswini
2015-08-22 08:31:27 
by using the above method i need the answer for
6752×111
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profilejay
2015-08-25 21:43:11 
6752×111

6 (6+7) (6+7+5) (7+5+2) (5+2) 2
6 3(c=1) 8(c=1) 4(c=1) 7 2
7 4 9 4 7 2

Ans is 749472
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profileJay
2014-06-08 10:50:51 
3rd examples answer is wrong as 1257*111=139527....so is der any fault in d method with 4 digit no or der is any other method 2 solve it
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profiles
2014-09-02 05:02:54 

Calculate is for  1257 * 1111 not  1257 * 111

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profilehemant
2014-06-11 16:38:03 
see here is another method

111*1257 --- add those numbers

1257+12570+125700 ---- add one zero from the second digit of 111

suppose 11111*1257 ---- add those numbers

1257+12570+125700+1257000+12570000 ---- add one zero from the second digit of 11111
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profileJay
2014-06-10 10:22:49 
3rd example is 1257 * 1111 = 1396527 which is correct
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profilesan
2014-03-07 03:47:31 
  1. 19385679*1111=?(how to solve
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profileJay
2014-03-07 20:24:28 
19385679*1111

1    1+9         1+9+3     1+9+3+8   9+3+8+5   3+8+5+6  8+5+6+7  5+6+7+9  6+7+9    7+9         9
1     10             13           21           25              22           26               27         22           16           9
1     (c=1) 0    (c=1) 3    (c=2)1    (c=2) 5       (c=2) 2     (c=2) 6       (c=2) 7  (c=2)2     (c=1) 6    9
1+1   0+1        3+2          1+2        5+2          2+2            6+2            7+2        2+1          6         9
2       1             5              3             7               4                8                9            3            6        9

ie 21537489369 is the answer
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