Question 11 Mark
What is the HCF of two consecutive odd numbers$?$
Answer
View full question & answer→$HCF$ of two consecutive odd numbers$ = 1$
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Question 21 Mark
What is the $HCF$ of two consecutive even numbers$?$
Answer
View full question & answer→$HCF$ of two consecutive even numbers $= 2$
Question 31 Mark
What is the $HCF$ of two consecutive numbers$?$
Answer
View full question & answer→$HCF$ of consecutive numbers $= 1$
Question 41 Mark
In the given expression, prime factorisation has been done or not$?\ 54 = 2 \times 3 \times 9$
Answer
View full question & answer→Given: $54 = 2 \times 3 \times 9$
As we can see in this factorization 9 is a composite number, and it can be further factorized, so prime factorization has not been done.
As we can see in this factorization 9 is a composite number, and it can be further factorized, so prime factorization has not been done.
Question 51 Mark
In the given expression, prime factorisation has been done or not$?\ 70 = 2 \times 5 \times 7$
Answer
View full question & answer→Given: $70 = 2 \times 5 \times 7$
Since all the factors are prime numbers, so, prime factorization has been done.
Since all the factors are prime numbers, so, prime factorization has been done.
Question 61 Mark
In the given expression, prime factorisation has been done or not? $56 = 7 \times 2 \times 2 \times 2$
Answer
View full question & answer→Given: $56 = 7 \times 2 \times 2 \times 2$
Since all the factors are prime, so, prime factorization has been done.
Since all the factors are prime, so, prime factorization has been done.
Question 71 Mark
In the given expressions, prime factorisation has been done or not? $24 = 2 \times 3 \times 4$
Answer
View full question & answer→Given that, $24 = 2 \times 3 \times 4$
As we can see in this factorization $4$ is a composite number, which can be further factorized, so prime factorization has not been done.
As we can see in this factorization $4$ is a composite number, which can be further factorized, so prime factorization has not been done.
Question 81 Mark
Which factors are not included in the prime factorisation of a composite number?
Answer
View full question & answer→$1$ and the number itself are not included in the prime factorisation of a composite number.
Question 91 Mark
I am the smallest number, having four different prime factors. Can you find me$?$
Answer
View full question & answer→Let us take the smallest four different prime numbers i.e $2, 3, 5$ and $7.$
$\therefore$ The smallest number, having four different prime factors is
$2 \times 3 \times 5 \times 7 = 210$
$\therefore$ The smallest number, having four different prime factors is
$2 \times 3 \times 5 \times 7 = 210$
Question 101 Mark
A number is divisible by $12.$ By what other numbers will that number be divisible?
Answer
View full question & answer→Factors of $12$ are $1, 2, 3, 4, 6$ and $12.$
Therefore the number will be divisible by $1, 2, 3, 4$ and $6.$
Therefore the number will be divisible by $1, 2, 3, 4$ and $6.$
Question 111 Mark
A number is divisible by both $5$ and $12.$ By which other number will that number be always divisible?
Answer
View full question & answer→The number will always be divisible by $5 \times 12 = 60.$
Question 121 Mark
Write the smallest digit and the largest digit in the blank space of number so that the number is divisible by $3 :\_\_\ 6724$
Answer
View full question & answer→$i.$ Smallest digit
Sum of the given digits $= 6 + 7 + 2 + 4 = 19$
$\because 19$ is not divisible by $3.$ The no. after $19$ which is divisible by $3$ is $21.$ So we will add $2$ to make it divisible by $3.$
$\therefore$ Smallest digit (non-zero) is $2.$
$ii.$ Largest digit
Sum of the given digits is $19.$ Largest no. which we can add is $9,$ so $19+9=28$ which is not divisible by $3$ hence $8$ should be added because $19+8=27$ which is divisible by $3.$
The largest digit is $8.$
Sum of the given digits $= 6 + 7 + 2 + 4 = 19$
$\because 19$ is not divisible by $3.$ The no. after $19$ which is divisible by $3$ is $21.$ So we will add $2$ to make it divisible by $3.$
$\therefore$ Smallest digit (non-zero) is $2.$
$ii.$ Largest digit
Sum of the given digits is $19.$ Largest no. which we can add is $9,$ so $19+9=28$ which is not divisible by $3$ hence $8$ should be added because $19+8=27$ which is divisible by $3.$
The largest digit is $8.$
Question 131 Mark
Using divisibility test, determine if $901153$ is divisible by $11.$
Answer
View full question & answer→Sum of the digits (at odd places) from the right $= 3 + 1 + 0 = 4$
Sum of the digits (at even places) from the right $= 5 + 1 + 9 = 15$
Difference of these sums $= 15 – 4 = 11$
$\because 11$ is divisible by $11.$
$\therefore 901153$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of digits at odd places and even places from the right is either $0$ or divisible by $11.$
Sum of the digits (at even places) from the right $= 5 + 1 + 9 = 15$
Difference of these sums $= 15 – 4 = 11$
$\because 11$ is divisible by $11.$
$\therefore 901153$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of digits at odd places and even places from the right is either $0$ or divisible by $11.$
Question 141 Mark
Using divisibility test, determine if $10000001$ is divisible by $11.$
Answer
View full question & answer→Sum of the digits (at odd places) from the right $= 1 + 0 + 0 + 0 = 1$
Sum of the digits (at even places) from the right $= 0 + 0 + 0 + 1 = 1$
Difference of these sums $= 1 – 1 = 0$
$\because 0$ is divisible by $11.$
$\therefore 10000001$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of the digits at the odd places and even places from the right is $0$ or divisible by $11.$
Sum of the digits (at even places) from the right $= 0 + 0 + 0 + 1 = 1$
Difference of these sums $= 1 – 1 = 0$
$\because 0$ is divisible by $11.$
$\therefore 10000001$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of the digits at the odd places and even places from the right is $0$ or divisible by $11.$
Question 151 Mark
Using divisibility tests, determine if $70169308$ is divisible by $11.$
Answer
View full question & answer→Sum of the digits (at odd places) from the right $= 8 + 3 + 6 + 0 = 17$
Sum of the digits (at even places) from the right $= 0 + 9 + 1 + 7 = 17$
Difference of these sums $= 17 – 17 = 0$
$\therefore 70169308$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of the digits at odd places and even places from the right is either $0$ or divisible by $11$
Sum of the digits (at even places) from the right $= 0 + 9 + 1 + 7 = 17$
Difference of these sums $= 17 – 17 = 0$
$\therefore 70169308$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of the digits at odd places and even places from the right is either $0$ or divisible by $11$
Question 161 Mark
Using divisibility test, determine if $7138965$ is divisible by $11.$
Answer
View full question & answer→Sum of the digits (at odd places) from the right $= 5 + 9 + 3 + 7 = 24$
Sum of the digits (at even places) from the right $= 6 + 8 + 1 = 15$
Difference of these sums $= 24 – 15 = 9$
$\because 9$ is not divisible by $11$
$\therefore 7138965$ is not divisible by $11$ because a no. is divisible by $11$ if difference of the sum of digits at odd places and even places from the right is either $0$ or divisible by $11.$
Sum of the digits (at even places) from the right $= 6 + 8 + 1 = 15$
Difference of these sums $= 24 – 15 = 9$
$\because 9$ is not divisible by $11$
$\therefore 7138965$ is not divisible by $11$ because a no. is divisible by $11$ if difference of the sum of digits at odd places and even places from the right is either $0$ or divisible by $11.$
Question 171 Mark
Using divisibility test, determine if $10824$ is divisible by $11.$
Answer
View full question & answer→Sum of the digits (at odd places) from the right $= 4 + 8 + 1 = 13$
Sum of the digits (at even places) from the right $= 2 + 0 = 2.$
Difference of these sums $= 13 – 2 = 11$
$\because 11$ is divisible by $11$
$\therefore 10824$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of the digits at odd places and even places from the right is either $0$ or divisible by $11.$
Sum of the digits (at even places) from the right $= 2 + 0 = 2.$
Difference of these sums $= 13 – 2 = 11$
$\because 11$ is divisible by $11$
$\therefore 10824$ is divisible by $11$ because a no. is divisible by $11$ if difference of the sum of the digits at odd places and even places from the right is either $0$ or divisible by $11.$
Question 181 Mark
Using divisibility tests, determine if $5445$ is divisible by $11.$
Answer
View full question & answer→Sum of the digits (at odd places) from the right $= 5 + 4 = 9$
Sum of the digits (at even places) from the right $= 4 + 5 = 9$
Difference of these sums $= 9 – 9 = 0$
$\therefore 5445$ is divisible by $11$ because a no. is divisible by $11$ if difference of sums of digits at odd places and even places ( from right) is either 0 or divisible by $11$
Sum of the digits (at even places) from the right $= 4 + 5 = 9$
Difference of these sums $= 9 – 9 = 0$
$\therefore 5445$ is divisible by $11$ because a no. is divisible by $11$ if difference of sums of digits at odd places and even places ( from right) is either 0 or divisible by $11$
Question 191 Mark
Using divisibility test, determine if $12583$ is divisible by $6.$
Answer
View full question & answer→A number is divisible by $6,$ if it is divisible by $2\ \& \ 3$ both
$i.$ Divisibility by $2$
$\because$ Unit's digit $= 3,$ which is not from the digits $0, 2, 4, 6$ or $8.$
$\therefore 12583$ is not divisible by $2.$
$\therefore12583$ is not divisible by $6.$
$i.$ Divisibility by $2$
$\because$ Unit's digit $= 3,$ which is not from the digits $0, 2, 4, 6$ or $8.$
$\therefore 12583$ is not divisible by $2.$
$\therefore12583$ is not divisible by $6.$
Question 201 Mark
Using divisibility tests, determine if number $61233$ is divisible by $6.$
Answer
View full question & answer→A number is divisible by $6,$ if it is divisible by $2\ \&\ 3$ both.
Divisibility by $2$
$\because$ Unit's digit $= 3,$ which is not any of the digits from $0, 2, 4, 6$ or $8.$
$\therefore 61233$ is not divisible by $2.$
$\therefore 61233$ is not divisible by $6$ because a no . is divisible by $6$ only if it is divisible by $2$ and $3$ both.
Divisibility by $2$
$\because$ Unit's digit $= 3,$ which is not any of the digits from $0, 2, 4, 6$ or $8.$
$\therefore 61233$ is not divisible by $2.$
$\therefore 61233$ is not divisible by $6$ because a no . is divisible by $6$ only if it is divisible by $2$ and $3$ both.
Question 211 Mark
Using divisibility tests, determine $17852$ is divisible by $6.$
Answer
View full question & answer→Sum of digits $= 1+7+8+5+2=23$
Which is not a multiple of $3.$
Therefore, $17852$ is not divisible by $3.$
Hence, it is not divisible by $6.$
Which is not a multiple of $3.$
Therefore, $17852$ is not divisible by $3.$
Hence, it is not divisible by $6.$
Question 221 Mark
Write seven consecutive composite numbers less than $100$ so that there is no prime number between them.
Answer
View full question & answer→Seven consecutive composite numbers less than $100$ are $90, 91, 92, 93, 94, 95, 96$
Question 231 Mark
Give three pairs of prime numbers whose difference is $2.$
$($Remark: Two prime numbers whose difference is $2$ are called twin primes$).$
$($Remark: Two prime numbers whose difference is $2$ are called twin primes$).$
Answer
View full question & answer→The three pairs of prime numbers, whose difference is $2,$ are as follows:
$3, 5; 5, 7; 11, 13.$
$3, 5; 5, 7; 11, 13.$
Question 241 Mark
Express the sum of two odd primes of $18$
Answer
View full question & answer→The number can be expressed as: $18 = 13 + 5$
Question 251 Mark
Express $24$ as the sum of two odd primes.
Answer
View full question & answer→The number can be expressed as: $24 = 17 + 7$
Question 261 Mark
Express the sum of two odd primes of $36.$
Answer
View full question & answer→The given number can be written as the sum of two odd primes as: $36 = 29 + 7$
Question 271 Mark
Express the sum of two odd primes of $44.$
Answer
View full question & answer→The given number can be expressed as $44 = 39 + 5$
Question 281 Mark
What is the greatest prime number between $1$ and $10?$
Answer
View full question & answer→The greatest prime number between $1$ and $10$ is $7.$
Question 291 Mark
Write down separately the prime and composite numbers less than $20.$
Answer
View full question & answer→Prime no. $= 2,3,5,7,11,13,17,19$
Composite no. $= 4,6,8,9,10,12,14,15,16,18$
Composite no. $= 4,6,8,9,10,12,14,15,16,18$
Question 301 Mark
The numbers $13$ and $31$ are prime numbers. Both these numbers have same digits $1$ and $3.$ Find such pairs of prime numbers upto $100.$
Answer
View full question & answer→All other such pairs of prime numbers upto $100$ are as follows.
$17$ and $71; 37$ and $73; 79$ and $97.$
$17$ and $71; 37$ and $73; 79$ and $97.$
Question 311 Mark
A number which has more than two factors is called a ________.
Answer
View full question & answer→composite number
Question 321 Mark
A number having only two factors is called a ________.
Answer
View full question & answer→prime number
Question 331 Mark
What is the sum of any two even numbers?
Answer
View full question & answer→The sum of two even numbers is even.
Question 341 Mark
Write five pairs of prime numbers below $20$ whose sum is divisible by $5.$Hint: $3 + 7 = 10$
Answer
View full question & answer→$2, 3 ; 3, 7 ; 2, 13 ; 3, 17; 7, 13.$
Question 351 Mark
What is the sum of any two odd numbers?
Answer
View full question & answer→The sum of two odd numbers is even.
Example: $3 + 5 = 8$
Example: $3 + 5 = 8$
Question 361 Mark
Express $61$ as the sum of three odd prime.
Answer
View full question & answer→Here,
$61 = 13 + 17 + 31$
Where, $13, 17,$ and $31$ are prime numbers.
$61 = 13 + 17 + 31$
Where, $13, 17,$ and $31$ are prime numbers.
Question 371 Mark
Express $53$ as the sum of three odd prime.
Answer
View full question & answer→Here,
$53 = 5 + 17 + 31$
Where, $5, 17,$ and $31$ are prime numbers.
$53 = 5 + 17 + 31$
Where, $5, 17,$ and $31$ are prime numbers.
Question 381 Mark
Express $31$ as the sum of three odd prime.
Answer
View full question & answer→Here,
$31 = 5 + 7 + 19$
Where, $3, 7,$ and $19$ are prime numbers.
$31 = 5 + 7 + 19$
Where, $3, 7,$ and $19$ are prime numbers.
Question 391 Mark
Express $21$ as the sum of three odd prime.
Answer
View full question & answer→$21$ can be written as,
$21 = 5 + 5 + 11$
where, $5$ and $11$ are prime numbers.
$21 = 5 + 5 + 11$
where, $5$ and $11$ are prime numbers.
Question 401 Mark
Write first five multiples of $5$
Answer
View full question & answer→The first five multiplies of $5$ can be obtained as follows;
$1 \times 5 = 5$
$2 \times 5 = 10$
$3 \times 5 = 15$
$4 \times 5 = 20$
$5 \times 5 = 25$
So, the first five multiplies are $5, 10, 15, 20$ and $25.$
$1 \times 5 = 5$
$2 \times 5 = 10$
$3 \times 5 = 15$
$4 \times 5 = 20$
$5 \times 5 = 25$
So, the first five multiplies are $5, 10, 15, 20$ and $25.$
Question 411 Mark
Write all the factors of $36$
Answer
View full question & answer→$36 = 1 \times 36$
$36 = 2 \times 18$
$36 = 3 \times 12$
$36 = 4 \times 9$
$36 = 6 \times 6$
Thus, all the factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18 $ and $36.$
$36 = 2 \times 18$
$36 = 3 \times 12$
$36 = 4 \times 9$
$36 = 6 \times 6$
Thus, all the factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18 $ and $36.$
Question 421 Mark
Write all the factors of $23$
Answer
View full question & answer→$23 = 1 \times 23$
As $23$ is a prime number so its factors are $1$ and $23$ only.
As $23$ is a prime number so its factors are $1$ and $23$ only.
Question 431 Mark
Write all the factors of $18$
Answer
View full question & answer→$18 = 1 \times 18$
$18 = 2 \times 9$
$18 = 3 \times 6$
Thus, all the factors of $18$ are $1, 2, 3, 6, 9$ and $18.$
$18 = 2 \times 9$
$18 = 3 \times 6$
Thus, all the factors of $18$ are $1, 2, 3, 6, 9$ and $18.$
Question 441 Mark
Write all the factors of $20$
Answer
View full question & answer→$20 = 1 \times 20$
$20 = 2 \times 10$
$20 = 4 \times 5$
Thus, all the factors of $20$ are $1, 2, 4, 5, 10$ and $20.$
$20 = 2 \times 10$
$20 = 4 \times 5$
Thus, all the factors of $20$ are $1, 2, 4, 5, 10$ and $20.$
Question 451 Mark
Write all the factors of $12$
Answer
$\therefore L.C.M. $ of $63, 70$ and $77 = 2 \times 3 \times 3 \times 5 \times 7 \times 11$
$= 6930$
Hence, the minimum distance each should cover so that all cover the distance in complete steps is $6930\ cm.$
View full question & answer→$\therefore L.C.M. $ of $63, 70$ and $77 = 2 \times 3 \times 3 \times 5 \times 7 \times 11$
$= 6930$
Hence, the minimum distance each should cover so that all cover the distance in complete steps is $6930\ cm.$
Question 461 Mark
Write all the factors of $27$
Answer
View full question & answer→$27 = 1 \times 27$
$27 = 3 \times 9$
Thus, all the factors of $27$ are $1, 3, 9 $ and $27.$
$27 = 3 \times 9$
Thus, all the factors of $27$ are $1, 3, 9 $ and $27.$
Question 471 Mark
Write all the factors of $21$
Answer
View full question & answer→$21 = 1 \times 21$
$21 = 3 \times 7$
Thus, all the factors of $21$ are $1, 3, 7$ and $21.$
$21 = 3 \times 7$
Thus, all the factors of $21$ are $1, 3, 7$ and $21.$
Question 481 Mark
Write all the factors of $15$
Answer
View full question & answer→$15 = 1 \times 15$
$15 = 3 \times 5$
Thus, all the factors of $15$ are $1, 3, 5$ and $15.$
$15 = 3 \times 5$
Thus, all the factors of $15$ are $1, 3, 5$ and $15.$
Question 491 Mark
Here are two different factor trees for $60.$ Write the missing numbers.
Answer
View full question & answer→The given number is: $24$
$1 \times 24 = 24$
$2 \times 12 = 24$
$3 \times 8 = 24$
$4 \times 6 = 24$
So, $1, 2, 3, 4, 6, 8, 12$ and $24$ are the factors of $24.$
$1 \times 24 = 24$
$2 \times 12 = 24$
$3 \times 8 = 24$
$4 \times 6 = 24$
So, $1, 2, 3, 4, 6, 8, 12$ and $24$ are the factors of $24.$