Prime factors of 1729 are -7,13,19.
Ascending order =7<13<19.
Relation = 7,13,19 differ by 6

Greatest four-digit number = 9999
9999 = 3 3 11 101

1 is the factors which are not especially included in the prime factorization of a composite number. When the number is divisible by 2 co-prime numbers then they are divisible by their product.
When two given numbers are divisible by a number, the sum is also divisible by number. When 2 given numbers are divisible by number then its difference is divisible by that number.

(a)
(b)

(a) False as 6 is divisible by 3, but not by 9.
(b) True, as 9 = 3 x 3 Therefore, if a number is divisible by 9, then it will also be divisible by 3.
(c) False as 30 is divisible by 3 and 6 both, but it is not divisible by 18.
(d) True as 9 x 10 = 90 Therefore, If a number is divisible by 9 and 10 both, then it will also be divisible by 90.
(e) False as 15 and 32 are co-primes and also...

prime factorizations of
16= 2×2×2×2
28= 2×2×7
38= 2×19

Since the number is divisible by 12, it will also be divisible by its factors i.e., 1, 2, 3, 4, 6, 12. Clearly, 1, 2, 3, 4, and 6 are numbers other than 12 by which this number is also divisible.

Factors of 5=1,5
Factors of 12=1,2,3,4,6,12
As the common factor of these numbers is 1, the given two numbers are coprime and the number will also be divisible by their product, i.e. 60, and the factors of 60=1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

(a) 18 and 35
Factors of 18=1,2,3,6,9,18
Factors of 35=1,5,7,35
Common factor =1
Therefore, the given two numbers are co-prime.
(b)15 and 37
Factors of 15=1,3,5,15
Factors of 37=1,37
Common factors =1
Therefore, the given two numbers are co-prime.
(c) 30 and 415
Factors of 30=1,2,3,5,6,10,15,30
Factors of 415=1,5,83,415
Common factors =1,5
As these numbers have a common factor...

Multiples of 3 = 3,6,9,12,15…
Multiples of 4 = 4,8,12,16,20…
Common multiples = 12,24,36,48,60,72,84,96

(a) 6 and 8
Multiple of 6=6,12,18,24,30…
Multiple of 8=8,16,24,32……
common multiples =24,48,72
(b) 12 and 18
Multiples of 12=12,24,36,78
Multiples of 18=18,36,54,72 3
common multiples = 36,72,108

(a) 4,8,12
Factors of 4=1,2,4
Factors of 8=1,2,4,8
Factors of 12=1,2,3,4,6,12
Common factors =1,2,4
(b) 5,15, and 25
Factors of 5=1,5
Factors of 15=1,3,5,15
Factors of 25=1,5,25
Common factors =1,5

(a) 20 and 28
Factors of 20=1,2,4,5,10,20
Factors of 28=1,2,4,7,14,28
Common factors =1,2,4
(b) 15 and 25
Factors of 15=1,3,5,15
Factors of 25=1,5,25
Common factors =1,5
(c) 35 and 50
Factors of 35=1,5,7,35
Factors of 50=1,2,5,10,25,50
Common factors =1,5
(d) 56 and 120
Factors of 56=1,2,4,7,8,14,28,56
Factors of 120=1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
Common factors =1,2,4,8

The common factors of the following are:
(a) 8, 20
Hence, the common factors are
(b) 9, 15
Hence, the common factors are

(a) 92 __ 389
Sum of odd digits = 9 + (blank space) + 8 = 17 + blank space
Sum of even digits = 2 + 3 + 9 = 14
As we know,
The number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is divisible by 11.
If we make the sum of odd digits = 25
then we will have difference = 25 - 14 = 11
which is divisible by 11.
To make the...

(a) _6724
Sum of the remaining digits = 19
To make the number divisible by 3, the sum of its digits should be divisible by 3. The smallest multiple of 3 which comes after 19 is 21.
Therefore,
smallest number = 21 - 19 - 2
Now, 2 + 3 + 3 = 8 If we put 8, then the sum of the digits will be 27 and as 27 is divisible by 3, the number will also be divisible by 3.
Therefore, the largest number is...

(a) 5445
Sum of the digits at odd places = 5 + 4 = 9 Sum of the digits at even places = 4 + 5 = 9 Difference = 9 - 9 = 0 As the difference between the sum of the digits at odd places and the sum of the digits at even places is O, therefore, 5445 is divisible by 11.
(b) 10824
Sum of the digits at odd places = 4 + 8 + 1 = 13 Sum of the digits at even places = 2 + 0 = 2 Difference = 13 - 2 = 11...

(a) 297144
Since the last digit Of the number is 4, it is divisible by 2. On adding all the digits of the number, the sum obtained is 27. Since 27 is divisible by 3, the given number is also divisible by 3. As the number is divisible by both 2 and 3, it is divisible by 6.
(b) 1258
Since the last digit of the number is 8, it is divisible by 2. On adding all the digits of the number, the sum...

A number with 3 or more digits is divisible by 4 if the number formed by its last two digits is divisible by 4.
A number with 3 or more digits is divisible by 8 if the number formed by its last three digits is divisible by 8.
a) 572
72 is divisible by 4, hence the number is divisible by 4.
The number is not divisible by 8.
b) 726352
52 is divisible by 4, hence the number is...

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