Number System Examples

Q 1 - Which of the following is a prime number?

A - 187

B - 811

C - 341

D - 437

Explanation

```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.
```

Q 2 - Which of the following is the output of 6894 x 99 ?

A - 685506

B - 682506

C - 683506

D - 684506

Explanation

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

Q 3 - Which of the following is the output of 685798 x 125 ?

A - 8224750

B - 8225750

C - 8225950

D - 8224760

Explanation

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

Q 4 - Which of the following is the output of 43986 x 625 ?

A - 27491450

B - 27491350

C - 27491250

D - 27491750

Explanation

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

Q 5 - Which of the following is the output of 869 x 738 + 869 x 262 ?

A - 262000

B - 738000

C - 969000

D - 869000

Explanation

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

Q 6 - Which of the following is the output of 936 x 587 - 936 x 487 ?

A - 93600

B - 58700

C - 48700

D - 100

Explanation

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

Q 7 - Which of the following is the output of 1496 x 1496 ?

A - 3338016

B - 2238016

C - 2248016

D - 2258016

Explanation

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

We've used following formula here:

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

Q 8 - Which of the following is the output of 1607 x 1607 ?

A - 2581449

B - 2583449

C - 2582449

D - 2584449

Explanation

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

We've used following formula here:

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

Q 9 - Which of the following is the output of 596 x 596 - 104 x 104 ?

A - 377700

B - 366600

C - 355500

D - 344400

Explanation

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

We've used following formula here:

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

Q 10 - Which of the following is the output of 57 x 57 + 43 x 43 + 2 x 57 x 43 ?

A - 10000

B - 5700

C - 4300

D - 1000

Explanation

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

We've used following formula here:

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

Q 11 - Which of the following is the output of 93 x 93 + 73 x 73 - 2 x 93 x 73 ?

A - 200

B - 400

C - 300

D - 100

Explanation

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

We've used following formula here:

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

Q 12 - Which of the following is the output of (578 x 578 x 578 + 432 x 432 x 432) / (578 x 578 - 578 x 432 + 432 x 432) ?

A - 2000

B - 4000

C - 3000

D - 1000

Explanation

```(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
```

We've used following formula here:

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

Q 13 - Which of the following is the output of (141 x 141 x 141 - 58 x 58 x 58) / (141 x 141 + 141 x 58 + 58 x 58) ?

A - 83

B - 100

C - 90

D - 73

Explanation

```(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
```

We've used following formula here:

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

Q 14 - Which of the following is the output of 213 x 213 + 187 x 187 ?

A - 50338

B - 80338

C - 90338

D - 70338

Explanation

```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
```

Q 15 - Which of the following is the output of ((637 + 478)2 - (637 - 478)2) /(637 x 478) ?

A - 4

B - 6

C - 8

D - 24

Explanation

```((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
```

We've used following formulae here:

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

Q 16 - Which of the following is the output of ((964 + 578)2 + (964 - 578)2) /(964 x 964 + 578 x 578) ?

A - 4

B - 6

C - 8

D - 2

Explanation

```((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
```

We've used following formulae here:

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

Q 17 - On dividing a number by 342, 47 is the remainder. What will be remainder if same number is divided by 18?

A - 11

B - 6

C - 8

D - 2

Explanation

```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.
```

We've used following formulae here:

```Dividend = (Divisor x Quotient) + Reminder
```

Q 18 - What will be unit digit in (3157)754?

A - 8

B - 9

C - 7

D - 6

Explanation

```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.
```

We've used following formulae here:

```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.
```

Q 19 - What will be unit digit in 658 x 539 x 436 x 312?

A - 8

B - 9

C - 4

D - 6

Explanation

```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.
```

Q 20 - What will be unit digit in 357 x 641 x 763?

A - 8

B - 9

C - 4

D - 6

Explanation

```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
```

We've used following formulae here:

```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.
```