Question

How many of the integers from 1 to 86 (inclusive) contain the digit 4 or have the digit sum divisible by 4?

Answer 1

it's like puzzle in number system.

Let's try to solve it,
integers between 1 to 86 contain the digit 4
e.g., 4 , 14, 24, 34 , 40, 41, 42 , 43 , 44 , 45, 46, 47, 48 , 49 , 54 , 64, 74, 84.
total number of integers contain the digit 4 = 18

now we have to find number of integers , have the digit sum divisible by 4.
we know, maximum value of digit sum in case of upto two digit number = (9 + 9) = 18
so, digit sum will be 4, 8, 12, 16 .
case1 :- when digit sum is 4.
possible integers : 13, 22, 31, {excluding integers contain digit 4}
so , number of integers , N1 = 3

case 2 :- when digit sum is 8.
possible integers : 17, 26, 35, 53, 62, 71, 80 {excluding integers contain digit 4}
so, number of integers , N2 = 7

case 3 :- when digit sum is 12.
possible integers : 39, 57, 66, 75 {excluding integers contain digit 4}
so, number of integers = 4

case 4 :- when digit sum is 16,
possible integers : 79
so, number or integer = 1

so, total number of integers = 18 + 3 + 7 + 4 + 1
= 33

Answer 2

Answer:

Step-by-step explanation:

Let's try to solve it,

integers between 1 to 86 contain the digit 4

e.g., 4 , 14, 24, 34 , 40, 41, 42 , 43 , 44 , 45, 46, 47, 48 , 49 , 54 , 64, 74, 84.

total number of integers contain the digit 4 = 18

now we have to find number of integers , have the digit sum divisible by 4.

we know, maximum value of digit sum in case of upto two digit number = (9 + 9) = 18

so, digit sum will be 4, 8, 12, 16 .

case1 :- when digit sum is 4.

possible integers : 13, 22, 31, {excluding integers contain digit 4}

so , number of integers , N1 = 3

case 2 :- when digit sum is 8.

possible integers : 17, 26, 35, 53, 62, 71, 80 {excluding integers contain digit 4}

so, number of integers , N2 = 7

case 3 :- when digit sum is 12.

possible integers : 39, 57, 66, 75 {excluding integers contain digit 4}

so, number of integers = 4

case 4 :- when digit sum is 16,

possible integers : 79

so, number or integer = 1

so, total number of integers = 18 + 3 + 7 + 4 + 1

= 33

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