Consecutive Number Multiplied Again

Consecutive Number Multiplied Again

Dr Himanshu Shekhar

Kids! As previous post entitled “Multiplication of Consecutive Numbers” got very good response (more than 100 in one day) from readers, I am encouraged to extend the topic and explore another quick, easy, universal and better method of multiplying consecutive numbers. By practice this can be perfected for mental mathematics.

As per new method illustrated in this post, the multiplication of consecutive numbers is carried out using partition, as discussed in many previous posts. The product of consecutive numbers will have three partitions:

• Left partition is multiplication of left digits
• Right partition has 1-digit and it is given by product of right digits.
• Central partition is product of left digit of bigger number and the sum of right side digits.
• Central partition is also allocated 1-digit only.
• If number of digits is more than 1 for right and central partition, they are carried over to left side partition.

This is explained with examples, as done in other posts, earlier.

26x27 = 2x2|2x(6+7)|6x7 = 4|26|42 = 4|26+4|2 = 4|30|2 = 4+3|0|2 = 702

93x94 = 9x9|9x(3+4)|3x4 = 81|63|12 = 81|63+1|2 = 81|64|2 = 81+6|4|2 = 8742

72x73 = 7x7|7x(2+3)|2x3 = 49|35|6 = 49+3|5|6 = 5256

68x69 = 6x6|6x(8+9)|8x9 = 36|102|72 = 36|102+7|2 = 36|109|2 = 36+10|9|2 = 4692

57x58 = 5x5|5x(7+8)|7x8 = 25|75|56 = 25|75+5|6 = 25|80|6 = 25+8|0|6 = 3306

49x50 = 4x5|5x(9+0)|9x0 = 20|45|0 = 20+4|5|0 = 2450

80x81 = 8x8|8x(0+1)|0x1 = 64|8|0 = 6480

Although the process is illustrated for explanation in written form, but the multiplication can be explored mentally, without writing. Some practice is needed for that. This method is independent on type of end digits available and there is no need to remember the unit place digit of the numbers, to be multiplied (as needed in previous post for similar operation)

Corollary 1: The process can be extended to product of any two 2-digit numbers, but some modification is needed in central partition.

Central partition is sum of cross-multiplication of digits of the number. The left digit of one number is multiplied to right digit of the other and vice-versa and then added.

29x43 = 2x4|2x3+9x4|9x3 = 8|6+36|27 = 8|42|27 = 8|42+2|7 = 8|44|7 = 8+4|4|7 = 1247

48x87 = 4x8|4x7+8x8|8x7 = 32|28+64|56 = 32|92|56 = 32|92+5|6 = 32|97|6 = 32+9|7|6 = 4176

73x94 = 7x9|7x4+3x9|3x4 = 63|28+27|12 = 63|55|12 = 63|55+1|2 = 63|56|2 = 63+5|6|2 = 6862

87x37 = 8x3|8x7+7x3|7x7 = 24|56+21|49 = 24|77|49 = 24|77+4|9 = 24|81|9 = 24+8|1|9 = 3219

Corollary 2: This process of multiplication can be extended to three digit numbers also. It must be kept in mind that for multiplication of three digit numbers, each multiplicand is partitioned into 1-digit right partition and 2-digit left partition before multiplication. The reverse of this should not be done. First some simple examples are illustrated.

123x124 = 12x12|12x(3+4)|3x4 = 144|84|12 = 144|84+1|2 = 144|85|2 = 144+8|5|2 = 15252.

117x118 = 11x11|11x(7+8)|7x8 = 121|165|56 = 121|165+5|6 = 121|170|6 = 121+17|0|6 = 13806

113x114 = 11x11|11x(3+4)|3x4 = 121|77|12 = 121|77+1|2 = 121|78|2 = 121+7|8|2 = 12882

Corollary 3: The process of multiplication for larger 3-digit numbers may need multiple application of partition in getting the product. Examples are given below to illustrate the point.

432x234 = 43x23|43x4+2x23|2x4 = 4x2|4x3+3x2|3x3| 172+46|8 = 8|18|9| 218|8 = 8+1|8|9| 218|8 = 989+21|8|8 = 101088

456x824 = 45x82|45x4+6x82|6x4 = 4x8|4x2+5x8|5x2| 180+492|24 = 32|48|10| 672+2|4 = 32+4|8+1|0| 674|4 = 3690+67 |4|4 = 375744

Kids! Enjoy the new methods in mathematics. It is all mental mathematics with minimum writing, but practice is the key to such expertise. Guardians and teachers may contribute in augmenting the posts for better acceptability by kids.

Dr Himanshu Shekhar