Theory of Numbers Test

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Theory of Numbers Test - 8

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Weekly Quiz Competition

Question 1
1 / -0

What is the highest factor of 1573, except the number itself?

A
11

B
13

C
121

D
143

Solution

1,573 = 11 × 143
= 11 × 11 × 13
So, highest factor = 11 × 13 = 143
Question 2
1 / -0

Find the last digit of (173)⁹⁹.

A
9

B
7

C
0

D
3

Solution

Use the concept of power cycle, i.e. unit digit repeats after every power which is a multiple of 4.
99 has 24 complete cycles of 4 and 3 as remainder.
So, unit digit of (173)⁹⁹ will be the unit digit of 3³, which is 7.
Question 3
1 / -0

What is the rightmost non-zero digit of 270²⁷⁰ + 130¹³⁰?

A
1

B
3

C
7

D
9

Solution

270²⁷⁰ will have 270 zeroes at the end and 130¹³⁰ will have 130 zeroes at the end.
When these two are added, the rightmost non-zero numeral will be the rightmost non-zero numeral of 130¹³⁰, i.e. the last numeral of 3¹³⁰.
Since the last numeral of the power 3 repeats after every 4^th power, so the last numeral of 3¹³⁰ is equal to the last numeral of 3², i.e. 9.
Question 4
1 / -0

What is the last digit of the result, if 6⁶³ is divided by 4²³?

A
2

B
4

C
6

D
9

Solution

= = = 2¹⁷ × 3⁶³
The last numeral of 2¹⁷ = 2 and the last numeral of 3⁶³ = 7
So, the required answer is the last numeral of 14 (2 × 7) = 4.
Question 5
1 / -0

The smallest positive number which leaves remainder 1 when divided by 3, 4, 5 or 7 is

A
85

B
106

C
141

D
421

Solution

The least number divisible by 3, 4, 5 and 7 = LCM of 3, 4, 5, 7 = 420.
Now, 420 is divisible by 3, 4, 5 and 7.
420 + 1 = 421 will always leave remainder 1, when divided by 3, 4, 5 or 7.
Required number = 420 + 1 = 421
Question 6
1 / -0

Find the numbers between 400 and 550, which when divided by 6, 8, or 9, leave 5 as the remainder in each case.

A
421 and 493

B
436 and 462

C
437 and 509

D
462 and 514

Solution

L.C.M. of 6, 8, and 9 is 72.
∴The required numbers = 72k + 5 (k is any natural number)
Put k = 6 and k = 7 to get the numbers between 400 and 550
∴Required numbers = 72 × 6 + 5 = 437 and 72 × 7 + 5 = 509
Question 7
1 / -0

x and y are distinct two-digit numbers. If their HCF is 14, what is the maximum possible value of x + y?

A
112

B
196

C
28

D
182

Solution

The numbers can be 14k₁ and 14k₂, where k₁ and k₂ do not have any common factor.
So, the possible two-digit numbers are 14 × 6 and 14 × 7.
∴ x + y = 84 + 98 = 182
Question 8
1 / -0

Find the number of factors of 5005.

A
15

B
16

C
8

D
12

Solution

Number of factors in 5005 can be found by using prime factorisation.
5005 = 5 × 7 × 11 × 13
Or 5¹ × 7¹ × 11¹ × 13¹
So, number of factors = 2 × 2 × 2 × 2 = 16
Hence, the total number of factors of 5005 = 16
Question 9
1 / -0

How many factors of 1440 are perfect squares?

A
9

B
10

C
36

D
6

Solution

1440 = 2⁵ × 3² × 5
To get the perfect square, we have to select one term from 2⁰, 2², 2⁴and one term from3⁰, 3²
and one term from 5⁰.
So, the answer is 3 × 2 × 1.
The answer is 6.
Question 10
1 / -0

How many numbers divisible by each of the numbers 21, 36 and 66 are there such that they are less than 10,000?

A
2

B
3

C
4

D
5

Solution

21 = 3 × 7
36 = 2²× 3²
66 = 2 × 3 × 11
LCM of 21, 36, 66 = 2 × 2 × 3 × 3 × 7 × 11 = 2772
Multiples of 2772 less than 10,000 are 2772, 5544 and 8316.
Question 11
1 / -0

M and N are two distinct natural numbers. The HCF and LCM of M and N are K and L, respectively. If A is also a natural number, then which of the following relations is not possible?

A
K × L = A

B
K × A = L

C
L × A = K

D
K × A = M

Solution

Let the numbers be 4 and 6. Then, their LCM and HCF are 12 and 2, respectively.
That is, LCM × Natural number can never be equal to HCF. All other cases are possible.
HCF of two distinct natural numbers can never be a multiple of their LCM.
Therefore, option 3 is the correct answer.
Question 12
1 / -0

If HCF of 8 numbers has to be found by the division method, then how many steps would be required?

A
8

B
7

C
6

D
4

Solution

For n numbers, we need (n –1) steps to find HCF by division method.
∴ 8 numbers will have (8 – 1) = 7 steps
Question 13
1 / -0

Which least number must be subtracted from 1856, so that the remainder is 4 when divided by 7, 12, 16?

A
137

B
1361

C
140

D
172

Solution

LCM of (7, 12, 16) = 336
If we divide 1856 by 336, then remainder is 176.
Since, it is given that remainder in this condition is 4.
Hence, the least number to be subtracted = (176 - 4) = 172
Question 14
1 / -0

Express the sum of repeating decimals and as a repeating decimal.

A

B

C

D

Solution

x = 0. + 0.0
= + =
Or x = = =
Question 15
1 / -0

Find the value of

A

B

C

D

Solution

Given expression is equal to = .
Question 16
1 / -0

x = 0.ababab…… and y = 0.baaa…….. If x = y, then what is the relation between a and b?

A
a = b

B
a + b = 1

C
a = 2b

D
Both (1) and (2)

Solution

x = and y =
x = y =
But ab = 10a + b and ba = 10b + a.
∴ = 100a + 10b = 99b + 11a a = b
Question 17
1 / -0

If N = 1000⁸ - 8, what is the sum of its digits?

A
200

B
207

C
208

D
209

Solution

N = 1000⁸ - 8 = 10²⁴ - 8
N = Number with twenty three 9s, followed by 2.
That is, N = 9999 ... 92
The sum of digits = (23 × 9) + 2 = 207 + 2 = 209
Question 18
1 / -0

If x = – 0.25, which of the following is the greatest?

A

B

C

D
2^{x + 1}

Solution

x = – 0.25
Option (1): = = = 1.33
Option (2): = = 0.8
Option (3): = = ≈ 0.87
Question 19
1 / -0

If 2^{x + y} = 4^{x – y}, then 3^x/y is

A
0

B
1

C
3

D
27

Solution

2^{x + y} = 2^{2(x – y)} x + y = 2x – 2y
x = 3y
∴ 3^x/y = 3³ = 27
Question 20
1 / -0

If ´ = 1. What is the value of x?

A

B

C

D
1

Solution

Given × = 1
´ = 1.
= 2⁹
∴ = 9
x =