+0  
 
0
64
3
avatar

Find the number of 10-digit numbers where the sum of the digits is divisible by 5.

 Jan 16, 2020
 #1
avatar
0

The answer is 4,000,000.  cool

 Jan 16, 2020
 #2
avatar
0

%%time
# Efficient Python 3 program to sum up the # 
# multiples of digit n in a range of 2 - 10-digit integers #
# E.G. "What is the total sum of all 3-digit integers that are
#  mutiples of 7?". Answer =70,336 #
  
# find the Sum of having n digit  
# and divisible by the number 
def totalSumDivisibleByNum(digit, number): 
  
    # compute the first and last term 
    firstnum = pow(10, digit - 1) 
    lastnum = pow(10, digit) 
  
    # first number which is divisible 
    # by given number 
    firstnum = (firstnum - firstnum % number)+number
    
    # last number which is divisible 
    # by given number 
    lastnum = (lastnum - lastnum % number ) 
    
    # total divisible number 
    count =int ((lastnum - firstnum) / number+1  ) 
    print("Total count =", f"{count:,d}")
    # return the total sum 
    return int(((lastnum + firstnum) * count) / 2) 
  
  
# Driver code 
digit =10 ; num = 5

print("Total Sum =", f"{totalSumDivisibleByNum(digit, num):,d}") 
  
Total count = 1,800,000,000
Total Sum = 9,900,000,002,700,000,000
Wall time: 0 ns

 Jan 16, 2020
 #3
avatar+24097 
+2

Find the number of 10-digit numbers where the sum of the digits is divisible by 5.

 

The first digit cannot be zero, so 9 ways, the next 8 digits can be anything, so there are 10 ways for each of them.
We can then only choose two digits as last digit to satisfy the condition.

 

sum = \(9\times 10^8 \times 2 = 1~ 800~ 000~ 000\)

 

\(\begin{array}{|c|c|} \hline \text{the sum of the digits 1 until 9} & \text{last digit to satisfy the condition }\\ \hline \ldots 0 & 0~ \text{ or } ~5 \\ \ldots 1 & 4~ \text{ or } ~9 \\ \ldots 2 & 3~ \text{ or } ~8 \\ \ldots 3 & 2~ \text{ or } ~7 \\ \ldots 4 & 1~ \text{ or } ~6 \\ \ldots 5 & 0~ \text{ or } ~5 \\ \ldots 6 & 4~ \text{ or } ~9 \\ \ldots 7 & 3~ \text{ or } ~8 \\ \ldots 8 & 2~ \text{ or } ~7 \\ \ldots 9 & 1~ \text{ or } ~6 \\ \hline \end{array} \)

 

Example: 1 000 000 08(1) or 1 000 000 08(6)

                4 567 123 98(0) or 4 567 123 98(5)

 

laugh

 Jan 17, 2020

23 Online Users