数制示例

答案 - B

解释

Step 1. Find a whole number k such that k2 > n for each number.
142 > 187.
302 > 811.
192 > 341.
212 > 437.

Step 2. Get all prime numbers which are < k
14 - 2 , 3, 5, 7, 11, 13
30 - 2 , 3, 5, 7, 11, 13, 17, 19, 23, 29
19 - 2 , 3, 5, 7, 11, 13, 17
21 - 2 , 3, 5, 7, 11, 13, 17, 19

Step 3. Check divisiblity of each number 
with prime numbers which are < k.
187 is divisible by 11.
811 is not divisible by any prime number.
341 is divisible by 11.
437 is divisible by 19.

Result: 811 is the prime number.

答案 - B

解释

6894 x 99
= 6894 x (100 - 1)
= 6894 x 100 - 6894 x 1
= 689400 - 6894
= 682506

答案 - A

解释

685798 x 125
= 685798 x 53
= 685798 x (10/2)3
= (685798 x 103) / 23
= 685798000 / 8
= 85724750

答案 - C

解释

43986 x 625
= 43986 x 54
= 43986 x (10/2)4
= (43986 x 104) / 24
= 439860000 / 16
= 27491250

答案 - D

解释

869 x 738 + 869 x 262
= 869 x (738 + 262)
= 869 x 1000
= 869000

答案 - A

解释

936 x 587 - 936 x 487
= 936 x (587 - 487)
= 936 x 100
= 93600

答案 - B

解释

1496 x 1496
= 14962
= (1500-4)2
= 15002 + 42 - 2 x 1500 x 4
= 2250000 + 16 - 12000
= 2238016

我们在这里使用了以下公式

(a-b)2 = a2 + b2 - 2ab.

答案 - C

解释

1607 x 1607
= 16072
= (1600+7)2
= 16002 + 72 + 2 x 1600 x 7
= 2560000 + 49 +22400
= 2582449

我们在这里使用了以下公式

(a+b)2 = a2 + b2 + 2ab.

答案 - D

解释

596 x 596 - 104 x 104
= 5962 - 1042
= (596 + 104) x (596 - 104)
= 700 x 492
= 344400

我们在这里使用了以下公式

a2 - b2 = (a + b)(a - b).

答案 - A

解释

57 x 57 + 43 x 43 + 2 x 57 x 43
= (57 + 43)2
= (100)2
= 10000

我们在这里使用了以下公式

(a + b)2 = a2 + b2 + 2ab.

答案 - B

解释

93 x 93 + 73 x 73 - 2 x 93 x 73
= (93 - 73)2
= (20)2
= 400

我们在这里使用了以下公式

(a - b)2 = a2 + b2 - 2ab.

答案 - D

解释

(578 x 578 x 578 + 432 x 432 x 432) / (578 x 578 - 578 x 432 + 432 x 432)
Let's have a = 578, b = 432

Now expression is (a3 + b3) / (a2 - ab + b2)
= a + b
= 578 + 432
= 1000

我们在这里使用了以下公式

a3 + b3  = (a + b)(a2 - ab + b2).

答案 - A

解释

(141 x 141 x 141 - 58 x 58 x 58) / (141 x 141 + 141 x 58 + 58 x 58)
Let's have a = 141, b = 58
Now expression is (a3 - b3) / (a2 + ab + b2)
= a - b
= 141 - 58
= 83

我们在这里使用了以下公式

a3 - b3  = (a - b)(a2 + ab + b2).

答案 - B

解释

213 x 213 + 187 x 187
Let's have a = 213, b = 187
Now expression is a2 + b2
Using following formula, (a + b)2 + (a - b)2 = 2 x (a2 + b2)
2 x ( 213 x 213 + 187 x 187) = (213 + 187)2 + (213 - 187)2
2 x ( 213 x 213 + 187 x 187) = 4002 + 262
2 x ( 213 x 213 + 187 x 187) = 160000 + 676
213 x 213 + 187 x 187 = 160676 / 2
= 80338

答案 - C

解释

((637 + 478)2 - (637 - 478)2)/(637 x 478)
Let's have a = 637, b = 478
Now expression is ((a + b)2 - (a - b)2) / ab
= (a2 + b2 + 2ab - (a2 + b2 - 2ab)) / ab
= (a2 + b2 + 2ab - a2 - b2 + 2ab) / ab
= 4ab / ab
= 4

我们在这里使用了以下公式

(a + b)2  = a2 + b2 + 2ab.
(a - b)2  = a2 + b2 - 2ab.

答案 - D

解释

((964 + 578)2 + (964 - 578)2) /(964 x 964 + 578 x 578)
Let's have a = 964, b = 578
Now expression is ((a + b)2 + (a - b)2) / (a2 + b2)
= (a2 + b2 + 2ab + (a2 + b2 - 2ab)) / (a2 + b2)
= (a2 + b2 + 2ab + a2 + b2 - 2ab) / (a2 + b2)
= 2(a2 + b2) / (a2 + b2)
= 2

我们在这里使用了以下公式

(a + b)2  = a2 + b2 + 2ab.
(a - b)2  = a2 + b2 - 2ab.

答案 - A

解释

Let's quotient is a and given number be b.
b = 342a + 47
= (18 x 19)a + 36 + 11
= (18 x 19)a + (18 x 2) + 11
= 18 x (19a + 2) + 11
Thus, if same number is divided by 18, remainder will be 11.

我们在这里使用了以下公式

Dividend = (Divisor x Quotient) + Reminder

答案 - B

解释

unit digit in (3157)754
= unit digit in (7)754
= unit digit in (74)188 x 72
= unit digit in (1 x 49) 
= 9
Thus Unit digit in (3157)754 is 9.

我们在这里使用了以下公式

Unit digit in 71 = 7
Unit digit in 72 = 9
Unit digit in 73 = 3
Unit digit in 74 = 1
Unit digit in 75 = 7
Unit digit in 76 = 9
Unit digit in 77 = 3
Unit digit in 78 = 1

So pattern is 7-9-3-1. This pattern works for all numbers. So Unit digit in ((7)4)n) will be 1.

答案 - C

解释

Multiply unit digits of each number.
Unit digit in 658 x 539 x 436 x 312 
= Unit digit in 8 x 9 x 6 x 2.
= Unit digit in 864.
= 4.

答案 - C

解释

357 = (34)14 x 3
So Unit digit in 357 
= Unit digit in 1 x 3 
= 3

641 = (64)10 x 6
So Unit digit in 641 
= Unit digit in 6 x 6 
= 6

763 = (74)15 x 73
So Unit digit in 761 
= Unit digit in 1 x 343 
= 3

So Unit digit in 357 x 641 x 763
= Unit digit in 3 x 6 x 3
= 4

我们在这里使用了以下公式

Unit digit in 34 = 1
Unit digit in 64 = 6
Unit digit in 74 = 1

So Unit digit 
- in ((3)4)n) will be 1.
- in ((6)4)n) will be 6.
- in ((7)4)n) will be 1.