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

P Pankaj Sanodiya
Smallest five-digit number = 10,000 10000 = 2 2 2 2 5 5 5 5

P Pankaj Sanodiya
Greatest four-digit number = 9999 9999 = 3  3 11 101

P Pankaj Sanodiya
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.

P Pankaj Sanodiya
(a) (b)

P Pankaj Sanodiya
(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...

P Pankaj Sanodiya
prime factorizations of 16= 2×2×2×2 28= 2×2×7 38= 2×19

P Pankaj Sanodiya
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.

P Pankaj Sanodiya
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.

P Pankaj Sanodiya
(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...

P Pankaj Sanodiya
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

P Pankaj Sanodiya
(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

P Pankaj Sanodiya
(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

P Pankaj Sanodiya
(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

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

P Pankaj Sanodiya
(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...

P Pankaj Sanodiya
(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...

P Pankaj Sanodiya
(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...