Question

Question#1
How many positive integers between 100 and 888 inclusive,

a) are divisible by 7?

b) are odd?

c) have distinct digits?

d) are not divisible by 6?

e) are divisible by either 4 or 7?

f) are not divisible by either 4 or 7?

g) are divisible by 4 but not by 7?

h) are divisible by 4 and 7?

Question#1
How many positive integers between 100 and 888 inclusive,

a) are divisible by 7?

b) are odd?

c) have distinct digits?

d) are not divisible by 6?

e) are divisible by either 4 or 7?

f) are not divisible by either 4 or 7?

g) are divisible by 4 but not by 7?

h) are divisible by 4 and 7?

please type the answers if possible! I've been having trouble reading the handwriting lately!

Answer 1

◆Here is your answer:

There are (888-100+1) integers including 100 and 888. So between 100 and 888, there are (888-100-1)=787 integers.

(a) Are divisible by 7

First integer grater than 100 and divisible by 7 is , 105=7×15.

And last integer divisible by 7 and less than 888 is , 882=7×126.

Therefore, number of integers divisible by 7 and lies between 100 and 888 is = (126-15+1)=112.

(b) are odd

first odd integer grater than 100 is 101 , and last odd integer less than 888 is 887.

Now, 101, 103, 105, 107,......., 885, 887 , this is a A.P. series. Whose first number is 101 and common difference is 2. Let 887 is nth number. Then ,

887 = 101 + (n-1)×2

or, 2(n - 1) = 786

or, n - 1 = 393

or, n = 394

Hence there are 394 ODD numbers between 100 and 888.

(c)Have distinct digits

Let the 3 digit number be like ABC.

Now if I take A = 1, then I have 8 number of choice (except 1) for B, and 7 number of choice (except 1, and B) for C. So total 8×7 number of choice in this case.

Similarly for, A = 2, 3, ...., 7 , we will get 8×7 numbers which have distinct digits.

Now for A=8 , B can't be 8 or 9, so number of choice for B is 7 . In this case C can't be 8 and B, so number of choice for C is also 7. So total 7×7 numbers are there in this case.

Combining all cases, there are , 7×(8×7) + 1×(7×7)=441 numbers, which have distinct digits.

(d) Are not divisible by 6

we see, 102 = 6×17

And, 882 = 6×147 , hence number of integers divisible by 6 and lies between 100 and 888 is = 147-17+1 = 131.

Total number of integers between 100 and 888 is 787.

So the number of integers between 100 and 888, not divisible by 6 is = 787 - 131 = 656

(e) Divisible by either 4 or 7

we see, 104 = 4 × 26

And, 884 = 4×221 , so number of integers between 100 and 888, divisible by 4 is = 221 - 26 +1 = 196

Also, from part (a) , number of integers between the 100 and 888 , divisible by 7 , is 112.

Now , 4×7 = 28

And, 112 = 28 × 4

868 = 28 × 31 , so the number of integers between 100 and 888 , divisible by both 4 and 7 is = ( 31 - 4 +1) = 28

Therefore number of integers between 100 and 888, divisible by 4 or 7 is = (196 + 112) - 28 = 280

(f) Are not divisi ble by either 4 or 7.

form part (e) number of integers between 100 and 888, divisible by 4 or 7 is 280.

Then number of integers not divisible by either 4 or 7 is =787 - 280 = 507

(g) Are divisible by 4 , but not by 7

from part (e) number of integers divisible by 4 is 196.

To exclude the numbers divisi ble by 7, we have to exclude the numbers divisible by 4 and 7 , i.e. by 28 from by only 4.

From part (e) number of integers divisible by 28 is, 28.

Therefore number of integers divisible by 4 but not by 7 is =196 - 28 = 168

(h) Are divisible by 4 and 7

from part (e) number of integer between 100 and 888 , and divisible by 4 and 7, i.e. by 28 is 28.

Thank you.