Multiplication of Consecutive Numbers

Multiplication of Consecutive Numbers

Dr Himanshu Shekhar

Kids! This time, I am exploring another aspect of Multiplication through examples, where method to find out product of consecutive integers will be illustrated.

In fact, in one of the post entitled “Multiplication of Numbers with Complimentary Digits”, such requirements are spelt out, which is reproduced below:

122x128 = 12x13 | 2x8 = 156 | 16 = 15616.

145x145 = 14x15 | 5x5 = 21025

The left partition invariably require, multiplication of two consecutive numbers. Rather than partitioning, as discussed in previous posts, the current post explores addition and some other properties based on last digits of the products. For ease of calculation, explanation using product of 2-digit consecutive numbers is discussed in the post, which can be extended further by practice.

The product of consecutive numbers, with unit place digit of one of the numbers as 0 or 5 can be obtained by direct multiplication, easily, as given below, with slight modification (extraction of 2&5 to give 2x5 = 10 from numbers).

89x90 = 8010; 
90x91 = 8190; 
94x95 = 47x19x10 = 8930; 
95x96 = 10x19x48 = 9120;

79x80 = 6320; 
80x81 = 6480; 
84x85 = 42x17x10 = 7140; 
85x86 = 10x17x43 = 7310;

The product of consecutive integers, where unit place digits of smaller number are 1 and 2, the product is obtained by addition of two parts:

• One part is obtained by product of equal separation of both consecutive numbers
• Second part is product of unit place digits.

Equal separation from consecutive numbers means either (i) 1 less than smaller and 1 more than larger number or (ii) 2 less than smaller and 2 more than larger number. This is illustrated by examples:

First examples with 1 less than smaller and 1 more than larger number is taken:

91x92 = 90x93+2 = 8370+2 = 8372;

81x82 = 80x83+2 = 6640+2 = 6642;

71x72 = 70x73+2 = 5110+2 = 5112;

61x62 = 60x63+2 = 3780+2 = 3782;

Examples with 2 less than smaller and 2 more than larger number is taken:

92x93 = 90x95+6 = 8550+6 = 8556;

82x83 = 80x85+6 = 6800+6 = 6806;

72x73 = 70x75+6 = 5250+6 = 5256;

62x63 = 60x65+6 = 3900+6 = 3906;

The product of consecutive integers, where unit place digits of the smaller number are 6 and 7, the product is obtained by addition of two parts:

• One part is obtained by product of equal separation of both consecutive numbers
• Second part is product of 5 less than each unit place digits.

Equal separation from consecutive numbers means either (i) 1 less than smaller and 1 more than larger number or (ii) 2 less than smaller and 2 more than larger number. This is illustrated by examples:

First examples with 1 less than smaller and 1 more than larger number is taken:

96x97 = 95x98+1x2 = 10x19x49+2 = 9310+2 = 9312;

86x87 = 85x88+1x2 = 10x17x44+2 = 7480+2 = 7482;

76x77 = 75x78+1x2 = 10x15x39+2 = 5850+2 = 5852;

66x67 = 65x68+1x2 = 10x13x34+2 = 4420+2 = 4422;

Examples with 2 less than smaller and 2 more than larger number is taken:

97x98 = 95x100+6 = 9500+6 = 9506;

87x88 = 85x90+6 = 7650+6 = 7656;

77x78 = 75x80+6 = 6000+6 = 6006;

67x68 = 65x70+6 = 4550+6 = 4556;

Now two type of consecutive number multiplication is left, where unit place digit of larger number is 4 or 9.

Main part of addition will be product of one less than smaller and one more than larger number and 2 is added to it. Examples are given below.

93x94 = 92x95+2 = 10x46x19+2 = 8740+2 = 8742;

98x99 = 97x100+2 = 9700+2 = 9702;

83x84 = 82x85+2 = 10x41x17+2 = 6970+2 = 6972;

88x89 = 87x90+2 = 7830+2 = 7832;

73x74 = 72x75+2 = 10x36x15+2 = 5400+2 = 5402;

78x79 = 77x80+2 = 6160+2 = 6162;

Now any set of multiplication of consecutive numbers can be carried out easily. It is just practice, which makes this method perfect. It is mental mathematics, which needs quick action by mind to get product of consecutive numbers.

Kids! Explore this method and compare with known conventional method. Enjoy.

Dr Himanshu Shekhar