2 Mark Question

Question 12 Marks

In the following numbers, replace * by the smallest number to make it divisible by 9:
835*86

Answer

835686
Here, 8 + 3 + 5 + * + 8 + 6 = 30 + * should be a multiple of 9.
To be divisible of 9, the least value of * should be 6, i.e., 30 + 6 = 36, which is a multiple of 9.
$\therefore$ * = 6

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Question 22 Marks

Test the divisibility of the following numbers by 6:
251780

Answer

A number is divisible by 6 if it is divisible by both 2 and 3.
Since 251780 is not divisible by 3, it is not divisible by 6.
Checking the divisibility by 3: The sum of the digits of the number, 2 + 5 + 1 + 7 + 8 + 0, is 23, which is not divisible by 3.
So, the number is not divisible by 3.

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Question 32 Marks

Test the divisibility of:
12030624 by 8

Answer

12030624 by 8
12030624 is divisible by 8.
It is because the number formed by its hundreds, tens and ones digits is 624, which is divisible by 8.

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Question 42 Marks

Find the HCF of the numbers in the following using the prime factorization method:504, 980

Answer

The given numbers are 504 and 980.We have:
$\begin{array}{c|c}2&504\\\hline2&252\\\hline2&126\\\hline3&63\\\hline3&21\\\hline7&7\\\hline&1\end{array}$
$\begin{array}{c|c}2&980\\\hline2&490\\\hline5&245\\\hline7&49\\\hline7&7\\\hline&1\end{array}$
$504=2\times2\times2\times3\times3\times7=2^3\times3^2\times7$
$980=2\times2\times5\times7\times7=2^2\times5\times7^2$
$\therefore$ HCF of the given numbers = $2^2$ × $7$ = $28$

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Question 52 Marks

Give the prime factorization of the following number:
1323

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}3&1323\\\hline3&441\\\hline3&147\\\hline7&49\\\hline7&7\\\hline&1\end{array}$
$\therefore1323=3\times3\times3\times7\times7\times1$
$=3^3\times7^2$

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Question 62 Marks

Give the prime factorization of the following number:
18

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}2&18\\\hline2&9\\\hline3&3\\\hline&1\end{array}$
$\therefore18=2\times3\times3$
$=2\times3^2$

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Question 72 Marks

Which of the following are prime numbers?
137

Answer

A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.
Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
137 is a prime number, because it is not divisible by 2, 3, 5, 7 and 11.

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Question 82 Marks

The product of two numbers is 2560 and their LCM is 320. Find their HCF.

Answer

Product of the two numbers = 2560.HCF = 320
We know that,
LCM × HCF = Product of two numbers
$\therefore$ HCF $=\frac{2560}{320}=8$

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Question 92 Marks

Test the divisibility of the following numbers by 7:
2345

Answer

To determine if a number is divisible by 7, double the last digit of the number and subtract it from the number formed by the remaining digits.
2345 is divisible by 7.
We have 234 - 2 × 5 = 224, which is a multiple of 7.

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Question 102 Marks

Which of the following are prime numbers?
331

Answer

A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.
Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
331 is a prime number, because it is not divisible by 2, 3, 5, 7, 11, 13, 17 and 19.

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Question 112 Marks

In the following numbers, replace * by the smallest number to make it divisible by 3:
8*711

Answer

81711
Here, 8 + * + 7 + 1 + 1 = 17 + * should be a multiple of 3.
To be divisible by 3, the least value of * should be 1, i.e., 17 + 1 = 18, which is a multiple of 3.
$\therefore$ * = 1

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Question 122 Marks

Find the LCM of the numbers given below:60, 75

Answer

The given numbers are 60 and 75.We have:
$\begin{array}{c|c}3&60,75\\\hline5&20,25\\\hline5&4,5\\\hline2&4,5\\\hline2&2,1\\\hline&1,1\end{array}$
$\therefore$ LCM = 3 × 5 × 5 × 2 × 2
= 300

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Question 132 Marks

Test the divisibility of the following numbers by 7:
6021

Answer

To determine if a number is divisible by 7, double the last digit of the number and subtract it from the number formed by the remaining digits.
6021 is divisible by 7.
We have 602 - 2 × 1 = 600, which is not a multiple of 7.

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Question 142 Marks

Write all prime numbers between 50 and 100.

Answer

53, 59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers between 50 and 100.

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Question 152 Marks

Which of the following are prime numbers?
161

Answer

A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.
Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
161 is a not prime number, because it is divisible by 7.

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Question 162 Marks

Show that the following pairs are co-primes:59, 97

Answer

The given numbers are 59 and 97.
59 = 59 × 1
97 = 97 × 1
$\therefore$ HCF = 1
Since 59 and 97 does not have any common factor other than 1, the two numbers are co-primes.

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Question 172 Marks

Which of the following are prime numbers?
397

Answer

A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.
Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
397 is a prime number, because it is not divisible by 2, 3, 5, 7, 11, 13, 17 and 19.

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Question 182 Marks

Test the divisibility of the following numbers by 7:
14126

Answer

To determine if a number is divisible by 7, double the last digit of the number and subtract it from the number formed by the remaining digits.
14126 is divisible by 7.
We have 1412 - 2 × 6 = 1400, which is a multiple of 7.

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Question 192 Marks

In the following numbers, replace * by the smallest number to make it divisible by 3:
6*1054

Answer

621054
Here, 6 + * + 1 + 0 + 5 + 4 = 16 + * should be a multiple of 3.
To be divisible by 3, the least value of * should be 2, i.e., 16 + 2 = 18, which is a multiple of 3.
$\therefore$ * = 2

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Question 202 Marks

Test the divisibility of:
10001001 by 3

Answer

10001001 by 3
10001001 is divisible by 3.
It is because the sum of its digits, 1 + 0 + 0 + 0 + 1 + 0 + 0 + 1, is 3, which is divisible by 3.

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Question 212 Marks

In the following numbers, replace * by the smallest number to make it divisible by 9:
6678*1

Answer

667881
Here, 6 + 6 + 7 + 8 + * + 1 = 28 + * should be a multiple of 9.
To be divisible by 9, the least value of * should be 8, i.e., 28 + 8 = 36, which is a multiple of 9.
$\therefore$ * = 8

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Question 222 Marks

What are composite numbers? Can a composite number be odd? If yes, write the smallest odd composite number.

Answer

COMPOSITE NUMBERS: Numbers having more than two factors are known as composite numbers.
Yes a composite number can odd. The smallest odd composite number is 9.

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Question 232 Marks

Find the LCM of the numbers given below:42, 63

Answer

The given numbers are 42 and 63.We have:
$\begin{array}{c|c}7&42,63\\\hline3&6,9\\\hline3&2,3\\\hline2&2,1\\\hline&1,1\end{array}$
$\therefore$ LCM = 7 × 3 × 3 × 2 × 1
= 126

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Question 242 Marks

Test the divisibility of the following numbers by 11:
66311

Answer

A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.
66311 is not divisible by 11.
Sum of the digits at odd places = (1 + 3 + 6) = 10
Sum of the digits at even places = (1 + 6) = 7
Difference of the two sums = (10 - 7) = 3, which is not divisible by 11.

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Question 252 Marks

Write seven consecutive composite numbers less than 100 having no prime number between them.

Answer

Seven consecutive composite numbers less than 100 having no prime number between them are 90, 91, 92, 93, 94, 95 and 96.

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Question 262 Marks

Give the prime factorization of the following number:
1035

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}3&1035\\\hline3&345\\\hline5&115\\\hline23&23\\\hline&1\end{array}$
$\therefore1035=3\times3\times5\times23$
$=3^2\times5\times23$

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Question 272 Marks

In the following numbers, replace * by the smallest number to make it divisible by 9:
2*135

Answer

27135
Here, 2 + * + 1 + 3 + 5 = 11 + * should be a multiple of 9.
To be divisible by 9, the least value of * should be 7, i.e., 11 + 7 = 18, which is a multiple of 9.
$\therefore$ * = 7

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Question 282 Marks

Test the divisibility of the following numbers by 11:
137269

Answer

A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.
137269 is divisible by 11.
Sum of the digits at odd places = (9 + 2 + 3) = 14
Sum of the digits at even places = (6 + 7 + 1) = 14
Difference of the two sums = (14 - 14) = 0, which is a divisible by 11.

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Question 292 Marks

Test the divisibility of the following numbers by 7:
826

Answer

To determine if a number is divisible by 7, double the last digit of the number and subtract it from the number formed by the remaining digits.
If their difference is a multiple of 7, the number is divisible by 7.
826 is divisible by 7.
We have 82 - 2 × 6 = 70, which is a multiple of 7.

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Question 302 Marks

In the following numbers, replace * by the smallest number to make it divisible by 3:
27*4

Answer

2724
Here, 2 + 7 + * + 4 = 13 + * should be a multiple of 3.
To be divisible by 3, the least value of * should be 2, i.e., 13 + 2 = 15, which is a multiple of 3.
$\therefore$ * = 2

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Question 312 Marks

Test the divisibility of:
1000001 by 11

Answer

10000001 by 11
10000001 is divisible by 11.
Sum of digits at odd places = (1 + 0 + 0 + 0) = 1
Sum of digits at even places = (0 + 0 + 0 + 1) = 1
Difference of the two sums = (1 - 1) = 0, which is divisible by 11.

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Question 322 Marks

Test the divisibility of the following numbers by 8:
901674

Answer

A number is divisible by 8 if the number formed by the last three digits (digits in the hundreds, tens and units places) is divisible by 8.
901674 is not divisible by 8.
It is because the number formed by its hundreds, tens and ones digits, i.e., 674, is not divisible by 8.

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Question 332 Marks

The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other.

Answer

HCF = 145LCM = 2175
One of the numbers = 725
We know that,
HCF × LCF = Product of two numbers
$\therefore$ Other number $=\frac{145\times2175}{725}=435$

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Question 342 Marks

Find the LCM of the numbers given below:36, 60, 72

Answer

The given numbers are 36, 60 and 72.We have:
$\begin{array}{c|c}2&36,60,72\\\hline2&18,30,36\\\hline3&9,15,18\\\hline3&3,5,6\\\hline5&1,5,2\\\hline2&1,1,2\\\hline&1,1,1\end{array}$
$\therefore$ LCM = 2 × 2 × 2 × 3 × 3 × 5
= 360

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Question 352 Marks

Give the prime factorization of the following number:
9317

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}7&9317\\\hline11&1331\\\hline11&121\\\hline11&11\\\hline&1\end{array}$
$\therefore9317=7\times11\times11\times11$
$=7\times11^3$

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Question 362 Marks

Test the divisibility of the following numbers by 8:
36792

Answer

A number is divisible by 8 if the number formed by the last three digits (digits in the hundreds, tens and units places) is divisible by 8.
36792 is divisible by 8.
It is because the number formed by its hundreds, tens and ones digits, i.e., 792, is divisible by 8

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Question 372 Marks

Test the divisibility of the following numbers by 8:
1790184

Answer

A number is divisible by 8 if the number formed by the last three digits (digits in the hundreds, tens and units places) is divisible by 8.
1790184 is divisible by 8.
It is because the number formed by its hundreds, tens and ones digits, i.e., 184, is divisible by 8.

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Question 382 Marks

Find the HCF of:2 and an even number.

Answer

2 and 4 are two prime numbers.
Now, HCF of 2 and 4 is as follows:
2 = 2 × 1
4 = 2 × 2 × 1
$\therefore$ HCF =2 × 1 = 2

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Question 392 Marks

Find the HCF of the numbers in the following using the division method:1965, 2096

Answer

The given numbers are 1965 and 2096.We have:

$\therefore$ The HFC is 131.

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Question 402 Marks

Which of the following are prime numbers?
217

Answer

A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
217 is a not prime number, because it is divisible by 7.

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Question 412 Marks

Find the HCF of the numbers in the following using the division method:2241, 2324

Answer

The given numbers are 2241 and 2341.
We have:

$\therefore$ The HFC = 83.

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Question 422 Marks

Test the divisibility of the following numbers by 8:
2138

Answer

A number is divisible by 8 if the number formed by the last three digits (digits in the hundreds, tens and units places) is divisible by 8.
2138 is not divisible by 8.
It is because the number formed by its hundreds, tens and ones digits, i.e., 138, is not divisible by 8.

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Question 432 Marks

Test the divisibility of the following numbers by 11:
4334

Answer

A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.
4334 is divisible by 11.
Sum of the digits at odd places = (4 + 3) = 7
Sum of the digits at even places = (3 + 4) = 7
Difference of the two sums = (7 - 7) = 0, which is divisible by 11.

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Question 442 Marks

Test the divisibility of the following numbers by 6:
872536

Answer

A number is divisible by 6 if it is divisible by both 2 and 3.Since 872536 is not divisible by 3, it is not divisible by 6.
Checking the divisibility by 3: The sum of the digits of the number, 8 + 7 + 2 + 5 + 3 + 6, is 31, which is not divisible by 3.
So, the number is not divisible by 3.

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Question 452 Marks

Give the prime factorization of the following number:
2907

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}3&2907\\\hline3&969\\\hline17&323\\\hline19&19\\\hline&1\end{array}$
$\therefore4641=3\times3\times17\times19$
$=3^2\times17\times19$

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Question 462 Marks

Test the divisibility of:
2134563 by 9

Answer

2134563 by 9
2134563 is not divisible by 9.
It is because the sum of its digits, 2 + 1 + 3 + 4 + 5 + 6 + 3, is 24, which is not divisible by 9.

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Question 472 Marks

Which of the following are prime numbers?
103

Answer

A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.
Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
103 is a prime number, because it is not divisible by 2, 3, 5, 7, 11 and 13.

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Question 482 Marks

In the following numbers, replace * by the smallest number to make it divisible by 3:
234*17

Answer

234117
Here, 2+ 3 +4 + * + 1 + 7 = 17 + * should be a multiple of 3.
To be divisible by 3, the least value of * should be 1, i.e., 17 + 1 = 18, which is a multiple of 3.
$\therefore$ * = 1

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Question 492 Marks

Give the prime factorization of the following number:
252

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}2&252\\\hline2&126\\\hline3&63\\\hline3&21\\\hline7&7\\\hline&1\end{array}$
$\therefore252=2\times2\times3\times3\times7\times1$
$=2^2\times3^2\times7\times1$

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Question 502 Marks

Give the prime factorization of the following number:
945

Answer

We will use the didvision method as shown below:
$\begin{array}{c|c}3&945\\\hline3&315\\\hline3&105\\\hline5&35\\\hline7&7\\\hline&1\end{array}$
$\therefore945=3\times3\times3\times5\times7\times1$
$=3^3\times5\times7$

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