How many three-digit numbers are divisible by 7?
By BYJU'S Exam Prep
Updated on: September 25th, 2023
A total of 128 three-digit numbers are divisible by 7. In this problem, we aimed to count the number of three-digit numbers that can be divided by 7 without leaving a remainder. By considering the range of three-digit numbers and identifying those that are divisible by 7, we can get the answer. Read further to know how many three-digit numbers are divisible by 7.
Table of content
Divisibility of 7
The divisibility of 7 helps us determine if a given number can be evenly divided by 7 without leaving a remainder. To check the divisibility of a number by 7, we can employ a simple rule. The rule states that if the alternating sum of the digits of a number is divisible by 7, then the number itself is also divisible by 7.
Divisibility of 7 Rule
- Start from the rightmost digit of the number and assign alternating positive and negative signs to the digits.
- Add up these signed digits.
- If the resulting sum is divisible by 7, then the original number is divisible by 7. Otherwise, it is not.
For example: let’s take the number 364.
Starting from the right, we have 4 – 6 + 3 = 1.
Since 1 is not divisible by 7, 364 is not divisible by 7.
Total Three-Digit Numbers Divisible by 7
We should find the number of three-digit numbers which are divisible by 7
The nth term of AP is aₙ = a + (n – 1)d
Where
aₙ is the nth term,
a is the first term,
d is a common difference,
n is the number of terms
First three-digit number divisible by 7 = 105
Next number = 105 + 7 = 112
So the series is 105, 112, 119, …
It forms an AP with 105 as the first term, and 7 is the common difference
If we divide 999 by 7, 5 will be the remainder
999 – 5 = 994 is the maximum possible three-digit number, which is divisible by 7
So the final sequence is 105, 112, 119, …. 994
the nth term of an AP = 994
a = 105
d = 7
aₙ = 994
We should find n
nth term of an A.P. is aₙ = a + (n – 1)d
994 = 105 + (n – 1)7
889 = (n – 1)7
n – 1 = 889/7
n – 1 = 127
n = 127 + 1
n = 128
Therefore, when asked How many three-digit numbers are divisible by 7? then the answer will be 128 three-digit numbers are divisible by 7.
Summary:
How many three-digit numbers are divisible by 7?
128 three-digit numbers are divisible by 7. The divisibility rule of 7 can be used to find these numbers. This rule is a helpful tool in solving mathematical problems and determining if a given number is divisible by 7.
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