Maths 1 Mark Question

False.
Solution:
2 and 3 are consecutive and are also prime numbers.

Rule: A natural number is divisible by 4 if the number formed by its last two digits is divisible by 4.
1020531
Here, the number formed by the last two digits is 31 which is not divisible by 4. Thus, 10,20,531 is not divisible by 4.

1 is neither composite nor prime.

Rule: A natural number is divisible by 2 if its unit digit is 0, 2, 4, 6 or 8.
984325
Here, the unit’s digit = 5
Thus, the given number is not divisible by 2.

The seven consecutive numbers, none of which is prime are:
90, 91, 92, 93, 94, 95, 96

The possible missing twins for 89 are 87 and 91. Since 87 and 91 are not primes, 89 has no twin.

5
In 55, the unit’s digit is 5. Hence, it is divisible by 5.

3. 2

Solution:

A number is divisible by 3 if the sum of its digits is divisible by 3.

The sum of digits in 6724 is 6 + 7+ 2 + 4 = 19

For divisible by 3 we have to add 2 in 19 i.e., 2 + 19 = 21, which is divisible by 3.

Hence, the correct answer is option (c).

70 and 90
The prime numbers between 70 and 90 are 71, 73, 79, 83 and 89.

A number which is divisible by 4 but not by 8 is 28.

A number which has only two factors is called a prime numbe.

False.Solution:
10 divides the sum of 18 and 2 (i.e., 20) but 10 divides neither 18 nor 2.

True.

We can express the given numbers as the sums of two odd primes as follows:
84 = 17 + 67 or 23 + 61

The smallest composite number is 4.

We can express the given numbers as the sums of two odd primes as follows:
36 = 7 + 29 or 17 + 19

False.Solution:
The even number 2 is not a composite number.

False.Solution:
2 + 3 = 5 which is a prime number.

The common factors of two consecutive even numbers are 1 and 2.
Therefore, HCF of two consecutive even numbers = 2

10 and 50
The prime numbers between 10 and 50 are 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

The possible twins for 101 are 99 and 103. Since 103 is a prime and 99 is not, 103 is the missing twin.

4. 98741032

Solution:

In 9,87,41,032, the number formed by the last three digits is 032, which is also divisible by 8.

Hence, 9,87,41,032 is divisible by 8.

60 and 100
The prime numbers between 60 and 100 are 61, 67, 71, 73, 79, 83, 89 and 97.

555
In 555, the unit’s digit is 5. Hence, it is divisible by 5.

True.

A number which is divisible by 4 and 8 but not by 32 is 48.

A number which is divisible by 3 but not by 6 is 9.

A number is divisible by 3 if the sum of its digits is divisible by 3.
607439
Here, the sum of the digits in the given number = 6 + 0 + 7 + 4 + 3 + 9 = 29 which is not divisible by 3.
Thus, 6,07,439 is not divisible by 3.

We can write the given numbers as the sums of the two or more primes as follows:
130 = 59 + 71

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
The given number = 7314
The number formed by its last three digits is 314 which is not divisible by 8.
Therefore, 7,314 is not divisible by 8.

Rule: A natural number is divisible by 4 if the number formed by its last two digits is divisible by 4.
786532
Here, the number formed by the last two digits is 32 which is divisible by 4. Thus, 7,86,532 is divisible by 4.

Rule: A number is divisible by 6 if it is divisible by 2 as well as 3.
732510
Here, the units digit = 0
Thus, the given number is divisible by 2.
Also, the sum of the digits = 7 + 3 + 2 + 5 + 1 + 0 = 18 which is divisible by 3. So, the given
number is divisible by 3.
Hence, 7,32,510 is divisible by 6.

We can write the given numbers as the sums of the two or more primes as follows:
180 = 139 + 17 + 11 + 13 or 79 + 101

A number having more than two factors is called a composite number.

False.Solution:
HCF of 6 and 9 is 3 not 1.

Rule: A natural number is divisible by 2 if its unit digit is 0, 2, 4, 6 or 8.
367314
Here, the unit’s digit = 4
Thus, the given number is divisible by 2.

True.

We can express the given numbers as the sums of twin primes which are as follows:
84 = 41 + 41

We can express the given numbers as the sums of twin primes which are as follows:
36 = 17 + 19

True.

False.
Solution:
12 is divisible by both 3 and 6 but it is not divisible by 18.

2. 1

Solution:

There is only one even number that is prime, i.e. 2.

The prime numbers between 50 and 100 are given below:
53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Rule: A natural number is divisible by 2 if its unit digit is 0, 2, 4, 6 or 8.
6520
Here, the unit’s digit = 0
Thus, the given number is divisible by 2.

We can express the given numbers as the sums of twin primes which are as follows:
120 = 59 + 61

We can express the given numbers as the sums of two odd primes as follows:
42 = 5 + 37 or 13 + 29

50005
In 50,005, the unit’s digit is 5. Hence, it is divisible by 5.

No, because each natural number is a factor of itself.

The smallest composite number is 4.Solution:
A composite number is a positive integer which is not prime (i.e., which has factors other than 1 and itself). Since, the factors of 4 are 1, 2, 4. Hence, the smallest composite number is 4.

The HCF of two consecutive odd numbers is 1.
Solution:
Since, the common factor in two consecutive odd numbers is only 1.
Hence, the HCF of two consecutive odd numbers is 1.

54 = 2 × 3 × 9 is not a prime factorization as 9 is not a prime number.

The least number which when divided by 5, 7 and 8 leaves 3 as remainder in each case is 283.
Solution:
LCM of 5, 7 and 8 is 5 × 7 × 8 = 280.
The least number which when divided by 5, 7 and 8 leaves 3 as remainder in each case is given by
280 + 3
= 283

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
The given number = 547218
The sum of the digits in the given number = 5 + 4 + 7 + 2 + 1 + 8 = 27 which is divisible by 9.
Therefore, 5,47,218 is divisible by 9.

A number is divisible by 3 if the sum of its digits is divisible by 3.
70335
Here, the sum of the digits in the given number = 7 + 0 + 3 + 3 + 5 = 18 which is divisible by 3.
Thus, 70,335 is divisible by 3.

We can express the given numbers as the sums of three odd prime numbers as follows:
31 = 5 + 7+ 9 + 13 or 31 = 11 + 13 + 7

The common factor of two consecutive odd numbers is 1.
Therefore, HCF of two consecutive odd numbers = 1

True.Solution:
The product of prime number is always a composite number.

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
The given number = 187245
The sum of the digits in the given number = 1 + 8 + 7 + 2 + 4 + 5 = 27 which is divisible by 9.
Therefore, 1,87,245 is divisible by 9.

False.
Solution:
20 is divisible by 4 but not by 8.

Rule: A natural number is divisible by 4 if the number formed by its last two digits is divisible by 4.
9801523
Here, the number formed by the last two digits is 23 which is not divisible by 4. Thus, 98,01,523 is not divisible by 4.

15625
15 is not a factor of 15,625 because it is not a divisor of 15,625.

40 and 85
The prime numbers between 40 and 85 are 41, 43, 47, 53, 59, 61, 67, 71, 73, 79 and 83.

False.
Solution:
9 is an odd number: 9 = 7 +2 where 7 and 2 are prime numbers.

False.Solution:
HCF of two consecutive odd numbers is 1. Example: HCF of 25 and 27 is 1.

False.
Solution:
9 is an odd number but it is composite numbers as its factor are 1, 3 and 9.

24 = 2 × 3 × 4 is not a prime factorization as 4 is not a prime number.

True.

56 = 1 × 7 × 2 × 2 × 2 is not a prime factorization as 1 is not a prime number.

We can write the given numbers as the sums of the two or more primes as follows:
13 = 11 + 2

True.
Solution:
A number and its successor have only one common factor (i.e., 1).

123015
15 is a factor of 1,23,015 because it is a divisor of 1,23,015. i.e., 8,201 × 15 = 1,23,015

We can replace the * by the smallest number to make the given numbers divisible by 3 as follows:
35 * 64
35 * 64 = 35064
As 3 + 5 +6 + 4 = 18, it is divisible by 3.

False.Solution:
12 is divisible by 3 but not by 9.

The prime triplet is (1, 5, 7).
Solution:
A set of three prime numbers which form an arithmetic sequence with common difference two is called a prime triplet.
Since, 3, 5 and 7 are satisfying the above condition.
Hence, the prime triplet is (3, 5, 7).

Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
The given number = 3478
The sum of the digits in the given number = 3 + 4 + 7 + 8 = 22 which is not divisible by 9.
Therefore, 3,478 is not divisible by 9.

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
The given number = 36712
Since the number formed by its last three digit = 712 which is divisible by 8.
Therefore, 36,712 is divisible by 8.

True.

70 = 2 × 5 × 7 is a prime factorization as 2, 5, and 7 are prime numbers.

A number which is divisible by 2 but not by 4 is 6.

We can express the given numbers as the sums of three odd prime numbers as follows:
49 = 3 + 5 + 41 or 49 = 7 + 11 +31

The smallest prime number is 2.

The required seven consecutive composite numbers are 90, 91, 92, 93, 94, 95 and 96.

False.
Solution:
4 and 9 are co-primes and both are composite numbers

True.

We can express the given numbers as the sums of three odd prime numbers as follows:
35 = 5 + 7 + 23 or 35 = 17 + 13 + 5

5555
In 5,555, the unit’s digit is 5. Hence, it is divisible by 5.

The LCM of two numbers is 26. The possible values of HCF are 1, 2 and 3.
Solution:
26 = 1 × 2 × 13
So, the possible values of HCF are 1, 2 and 13.

We can replace the * by the smallest number to make the given numbers divisible by 3 as follows:
75 * 5
75 * 5 = 7515
As 7 + 5 + 1 + 5 = 18, it is divisible by 3.

True.

two perfect numbers are 6 and 28.
Solution:
A perfect number is a positive number that equals the sum of its divisors, excluding itself.
Divisors of 6 = 1, 2, 3
Divisors of 28 = 1, 2, 4, 7, 14, 28
In 6, the sum of divisors except itself, 1 + 2 + 3 is 6.
In 28, the sum of divisors except itself, 1 + 2 + 4 + 7 + 14 is 28.
Therefore, two perfect numbers are 6 and 28.

The possible missing twins for 29 are 27 and 31. Since 31 is a prime and 37 is not, 31 is the missing twin.

The common factor of two consecutive numbers is always 1.
Therefore, HCF of two consecutive numbers = 1

Two perfect numbers are 6 and 28.
Solution:
A perfect number is a positive number that equals the sum of its divisors, excluding itself.
Divisors of 6 = 1, 2, 3
Divisors of 28 = 1, 2, 4, 7, 14, 28
In 6, the sum of divisors except itself, 1 + 2 + 3 is 6.
In 28, the sum of divisors except itself, 1 + 2 + 4 + 7 + 14 is 28.
Therefore, two perfect numbers are 6 and 28.

A number having only two factors is called a prime number.

Rule: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
The given number = 8364
The number formed by its last three digits is 364 which is not divisible by 8.
Therefore, 8,364 is not divisible by 8.

A number is divisible by 3 if the sum of its digits is divisible by 3.
9082746
Here, the sum of the digits in the given number = 9 + 0 + 8 + 2 + 7 + 4 + 6 = 36 which is divisible by 3.
Thus, 90,82,746 is divisible by 3.

We can replace the * by the smallest number to make the given numbers divisible by 3 as follows:
18 * 71
18 * 71 = 18171
As 1 + 8 + 1 + 7 + 1 = 18, it is divisible by 3.