Question

How many positive integers less than 1000 a) are divisible by 7? b) are divisible by 7 but not by 11? c) are divisible by both 7 and 11? d) are divisible by either 7 or 11? e) are divisible by exactly one of 7 and 11? f) are divisible by neither 7 nor 11? g) have distinct digits? h) have distinct digits and are even?

Answer 1

a)  7,14,28,35.........994 are 142 positive integer are divisible by 7
b)  77,154,..........924   are  12 positive integer are divisible by LCM of 11 or 7
so we can find only divisible by 7 not 11,
Subtract positive integer divisible by 7 - positive integer divisible by LCM of 11or 7=130 are divisible by 7 not by 11.c)
positive integer divisible by 7 is 142 and
positive integer divisible by 11 is 90
so add them and 232 positive integer is divisible by both 7 and 11.d)which is LCM of 11or 7 which is 77
so 132 positive integer are divisible by either 7or 11 .e)is it 232 positive ineteger is divisible by one of 7 and 11.f)divisible by 7 and 11 is 232 and total no is 1000 so subtract both768 positive integer is divisible by neither 7 nor 11.e) 500 distinct digit is are even

Answer 2

Given:

Set of positive integers less than 1000.

To Find:

a) are divisible by 7?

b) are divisible by 7 but not by 11?

c) are divisible by both 7 and 11?

d) are divisible by either 7 or 11?

e) are divisible by exactly one of 7 and 11?

f) are divisible by neither 7 nor 11?

g) have distinct digits?

h) have distinct digits and are even?

Solution:

a)

• Positive integers divisible by 7 below 1000 are 7,14,21,28,...,994.
• Total number of terms = 994 = 7 + (n-1)7 ( General form of an arithmetic progression).
• The total number of terms divisible by 7 below 1000 = 142.

b)

• The Numbers which are divisible by both 7 and 11 are 77, 154, 231, 308, 385, 462, 539, 616, 693, 770, 847, 924.
• Therefore, the total number of integers below 1000 which are divisible by 7 but not 11 = 142 - (total number of integers divisible by 7 and 11),

=> Total number of integers divisible by 7 but not 11 = 142-12 = 130.

c)

• Total number of integers which are divisible by both 7 and 11 = Total number of integers which are divisible by 77.
• The Total number of integers which are divisible by 77 below 1000 = 12.

d)

• Total number of integers which are either divisible by 7 or 11 = Total number of integers divisible by 7 + Total number of integers divisible by 11 - Total number of integers divisible by 77.

=> The total number of integers divisible by 11 below 1000, (11,22,33,...,990)

=> 990 = 11 + (n-1) 11,

=> 90 integers.

• Total number of integers which are divisible by both 7 and 11 = 142 + 90 - 12 = 220 integers.

e)

• Total number of integers that are exactly divisible by one of 7 and 11 = 142 + 90 - 12 = 220 integers.

f)

• Total number of integers that are neither divisible by 7 nor 11 = 1000 - (Total number of integers divisible by 7 or 11),

=> 1000 - 220 = 780 integers.

g)

• Total number of integers below 1000 having distinct digits = 1000 - (non-distinct digits),

=> Distinct digits  from 1 - 100 = 100 - ( 11,22,33,44,55,66,77,88,99,100),

=> Distinct digits from 1-100 = 90.

=> Distinct digits from 101-200 = 100 - (101,110,111,112,113,114,115,116,117,118,119,121,122,131,133,141,144,151,155,161,166,171,177,181,188,191,199,200),

=> Distinct digits from 101-200 = 100-28 = 72,

=> Distinct digits from 201 - 1000 = 72 x 8 = 576.

h)

• Distinct digits and even = 337 values . ( as the units digit is even the distinct digits reduces than the original value).