The quotients A = 2046 . Is and B = 3c8d : 18 are natural numbers. Find the sum of the smallest posible. A and the largest possible B
Answers
Step-by-step explanation:
The natural (or counting) numbers are 1,2,3,4,5,1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,...}{1,2,3,4,5,...}, is sometimes written NN for short.
The whole numbers are the natural numbers together with 00.
(Note: a few textbooks disagree and say the natural numbers include 00.)
The sum of any two natural numbers is also a natural number (for example, 4+2000=20044+2000=2004), and the product of any two natural numbers is a natural number (4×2000=80004×2000=8000). This is not true for subtraction and division, though.
Given:
The quotients A = 2a4b:15 and b = 3c8d:18 are natural numbers.
To Find:
Addition of the smallest possible value of A and largest possible value of B.
Solution:
The given problem can be solved using the divisibility concepts.
1. It is given that the value of 2a4b: 15 = A is a natural number, The least value of A is to be found,
=> For a number to be divisible by 15, It should be divisible by both 3 and 5.
=> For a number to be divisible by 3, the number digits sum must be divisible by 3.
=> For a number to be divisible by 5, the last digit must be either 0 or 5.
2. A = 2a4b: 15, the value of A is the smallest if the number 2a4b is small,
=> The value of b can be either 0 or 5 to be divisible by 5. Since A should be small the value of b is 0.
=> 2 + a + 4 + 0 = 6 + a must be divisible by 3, For a =0, the number 2040 can be divisible by 15 and it has the least possible value of A,
=> A = 2040/15 = 136.
Therefore, the value of A is 136.
3. B = 3c8d : 18 is a natural number,
=> For a number to be divisible by 18, It should be divisible by both 2 and 9.
=> For a number to be divisible by 2, the last digit must be even.
=> For a number to be divisible by 9, the digit sum must be divisible by 9.
=> 3c8d is divisible by 2 if d = 0,2,4,6,8. (Since the value of B is to be made as large as possible, the maximum possible value is taken).
=> d = 8,
=> 3 + c + 8 + 8 must be divisible by 9,
=> 19 + c must be divisible by 9,
=> The maximum value of c for which the number is divisible by 9 is c =9,
=> 3888 is the maximum possible number.
4. B = 3888/9 = 432. Therefore, the value of B is 432.
5. The sum of the smallest possible value of A and the largest possible value of B is 136+432 = 568.
Therefore, the sum of the values of A and B is 568.