Note that the figures on the tenth digit of the two factors on the left of each formula are the same, and the sum of the figures on the digits is 10. Find out the rules in the above three formulas and recalculate:

( 1)22×28; (2)24×26; (3)33×37; (4)45×45.

Usually, we use the multiplicative distribution rate for simple operations:

2 1×29=2 1×(30- 1)=2 1×30-2 1× 1=630-2 1=609;

23×27=23×(30-3)=23×30-23×3=690-69=62 1;

25×25=25×(30-5)=25×30-25×5=750- 125=625.

Or:

2 1×29=(20+ 1)×29=20×29+ 1×29=580+29=609;

23×27=(20+3)×27=20×27+3×27=540+8 1=62 1;

25×25=(20+5)×25=20×25+5×25=500+ 125=625.

In addition, according to the characteristics of the data (the sum of digits is 10), we can divide the data as follows, and then use the allocation rate to calculate:

2 1×29=(20+ 1)×(30- 1)=20×30-20× 1+ 1×30- 1× 1=609;

23×27=(20+3)×(30-3)=20×30-20×3+3×30-3×3=62 1;

25×25=(20+5)×(30-5)=20×30-20×5+5×30-5×5=625.

All the above methods can achieve the purpose of simple operation, but they do not achieve the purpose of fast calculation. If we analyze the figures in the unit and the decile respectively, it is not difficult to find that when the figures in the decile and the digit in the unit are the same and the sum of the figures in the unit is 10, their product is: the product of the digit in the decile and the number larger than it is 1 0, and then.

2 1×29=2×(2+ 1)× 100+ 1×9=600+9=609;

23×27=2×(2+ 1)× 100+3×7=600+2 1=62 1;

25×25=2×(2+ 1)× 100+5×5=600+25=625.

So, what is the basis of this calculation?

Let's assume that the number on the tenth digit of these two digits is A, and the number on the single digit of one digit is B, then the number on the single digit of the other digit is (10-b). So their products are:

( 10a+b)[ 10a+( 10-b)]= 100 a2+ 10a( 10-b)+ 10a b+b( 10-b)

= 100 a2+ 100 a- 10ab+ 10ab+b( 10-b)

= 100 a(a+ 1)+b( 10-b)

We also know that the product of two digits is less than 100. Therefore, we can write the product of two digits with the same number in ten digits and the sum of the numbers in one digit is 10 from high to low as: the product of the number in ten digits multiplied by the number greater than 1, and the product of the number in two digits.

For example: 2 1× 29: 2× (2+ 1) = 6, 1× 9 = 9.

Then there is 21× 29 = 609; (Pay attention to the digits of the product)

23×27:2×(2+ 1)=6,3×7=2 1

Then it is 23× 27 = 621;

25×25:2×(2+ 1)=6,5×5=25

Then it is 25×25=625.

( 1)22×28=6 16 2)24×26=624 3)33×37= 122 1

(4)45×45=2025 5) 1 1× 19=209 (6)76×74=5624

(7)120×180 = 21600 (pay attention to the number of digits of the product) (8)270×230=62 100.

This rule can be generalized: if the digits of two factors are the same in the multiplication operation, except that the sum of digits on one digit (set as b) is 10, and the digits on other digits (set as a) are exactly the same, then the product is:100a (a+1)+b (/kloc-).

Such as121×129:12× (13+1) =156, 1×9=9.

Then there is12/kloc-0 /×129 =15609;

253×257:25×26=650,3×7=2 1

Then it is 253× 257 = 65021;

3728×3722:372×373= 138756,8×2= 16

Then it is 3728× 3722 =13875616.

The product of two numbers that meet the conditions can be calculated quickly.

Using this rule, we can quickly calculate the square of any number with a unit number of 5. Any number with a unit number of 5 can be written as 10n+5, so (10n+5) 2 =100n (n+1)+25.

For example, 152=225, 252 = 625,352 =1221,

452=2025, 552=3025, 652=4225,

752=5625, 852=7225, 952=9025.

3752= 140625, 7952=632025, 1 1652= 1357225.