What is the biggest 3 digit number divisible by 7?

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From 1 to 100, if you want to find number of numbers which are divisible by 7,  how will you find?

Step 1 :

Divide 100 by 7

100/7 = 14.29

Step 1 :

When we divide 100 by 7, we get 14.29.

We should not round this result to the next integer or previous integer (that is 15 or 14).

We have to consider the largest integer contained in 14.29.

That is 14.

From 1 to 100, number of numbers divisible by 7 is 14.

Be careful! the above mentioned method will work only if you take the numbers from 1.

Note :

Whenever we want to find number of numbers which are divisible by k from 1 to a particular number x, we have to divide x by k.

To get the number of numbers divisible by k from 1 to x, we have to consider the largest integer contained in the result of x/k

In the result of x/k, we should not round to the nearest/next integer. For example, we should not round 14.9 to 15.

Question :

How many three digit numbers  are divisible by 7?

Answer :

Step 1 :

The smallest three digit number is 100.

The largest three digit number is 999.

Step 2 :

Let us find number of numbers divisible by 7 from 1 to 999.

Divide 999 by 7 ----> 999/7 = 142.71

Therefore number of numbers divisible by 7 from 1 to 999 is 142.

Step 3 :

Let us find number of numbers divisible by 7 from 1 to 99.

Divide 99 by 7 ----> 99/7 = 14.14.

Therefore number of numbers divisible by 7 from 1 to 99 is 14.

Step 4 :

Number of numbers divisible by 7 (1 to 999) = 142.

Number of numbers divisible by 7 (1 to 99) = 14.

Number of numbers divisible by 7 (100 to 999) :

= 142 - 14

= 128

Therefore, number of three digit numbers (100 to 999) which are divisible by 7 is 128.

This is how to we have to find number of 3 digit numbers divisible by 7.

The concept explained in the above problem will not only be helpful to find number of 3 digit numbers divisible by 7, it can also be applied to find number of 3 digit numbers which are divisible by any n umber.

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Solution:

We have to find the number of three-digit numbers which are divisible by 7.

aₙ = a + (n - 1)d is the nth term of AP.

Here, aₙ is the nth term, a is the first term, d is the common difference and n is the number of terms.

First three-digit number that is divisible by 7 = 105

Next number = 105 + 7 = 112

Therefore, the series becomes 105, 112, 119, ...

Thus, 105, 112, 119, ... is forming an A.P. having the first term as 105 and a common difference of 7.

When we divide 999 by 7, the remainder will be 5.

Clearly, 999 − 5 = 994 is the maximum possible three-digit number that is divisible by 7.

Hence the final sequence is as follows:

105, 112, 119, ..., 994

Let 994 be the nth term of this A.P.

a = 105

d = 7

aₙ = 994

n = ?

We know that the 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

There are 128 three-digit numbers that are divisible by 7

☛ Check: NCERT Solutions for Class 10 Maths Chapter 5

Video Solution:

NCERT Solutions Class 10 Maths Chapter 5 Exercise 5.2 Question 13

Summary:

The number of three-digit numbers divisible by 7 is 128.

☛ Related Questions:

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• Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12.
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Three digit numbers divisible by 7

We will start off with a couple of definitions for clarification. Three digit numbers (3-digit numbers) are numbers that have three digits in them. They range from 100 to 999. Therefore, there are a total of 900 3-digit numbers. Furthermore, a three digit number is divisible by 7 if you divide the three digit number by 7 and you get a whole number with no remainder. Below you will find many questions and answers related to three digit numbers divisible by 7.

How many three digit numbers are divisible by 7?

Below is a list of all three digit numbers divisible by 7 in chronological order. There are 128 three digit numbers divisible by 7. 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378, 385, 392, 399, 406, 413, 420, 427, 434, 441, 448, 455, 462, 469, 476, 483, 490, 497, 504, 511, 518, 525, 532, 539, 546, 553, 560, 567, 574, 581, 588, 595, 602, 609, 616, 623, 630, 637, 644, 651, 658, 665, 672, 679, 686, 693, 700, 707, 714, 721, 728, 735, 742, 749, 756, 763, 770, 777, 784, 791, 798, 805, 812, 819, 826, 833, 840, 847, 854, 861, 868, 875, 882, 889, 896, 903, 910, 917, 924, 931, 938, 945, 952, 959, 966, 973, 980, 987, 994
How many three digit numbers are NOT divisible by 7? Like we wrote above, there are a total of 900 three digit numbers and 128 of them are divisible by 7. Furthermore, 900 - 128 = 772. Therefore, there are 772 three digit numbers that are not divisible by 7.

What is the sum of all three digit numbers divisible by 7?

We added up all the 3-digit numbers in our list above. The sum of all 3-digit numbers divisible by 7 is 70336.

What is the smallest three digit number divisible by 7?

The smallest or lowest 3-digit number divisible by 7 is the first number on the list above (first 3 digit number divisible by 7). As you can see, that number is 105.

What is the largest three digit number divisible by 7?

The largest or greatest 3-digit number divisible by 7 is the last number on the list above (last 3 digit number divisible by 7). As you can see, that number is 994.
How many even three digit numbers are divisible by 7? Below is a list of all 3-digit EVEN numbers divisible by 7 in chronological order. There are 64 even three digit numbers divisible by 7. 112, 126, 140, 154, 168, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 378, 392, 406, 420, 434, 448, 462, 476, 490, 504, 518, 532, 546, 560, 574, 588, 602, 616, 630, 644, 658, 672, 686, 700, 714, 728, 742, 756, 770, 784, 798, 812, 826, 840, 854, 868, 882, 896, 910, 924, 938, 952, 966, 980, 994

How many odd three digit numbers are divisible by 7?

Below is a list of all 3-digit ODD numbers divisible by 7 in chronological order. There are 64 odd three digit numbers divisible by 7. 105, 119, 133, 147, 161, 175, 189, 203, 217, 231, 245, 259, 273, 287, 301, 315, 329, 343, 357, 371, 385, 399, 413, 427, 441, 455, 469, 483, 497, 511, 525, 539, 553, 567, 581, 595, 609, 623, 637, 651, 665, 679, 693, 707, 721, 735, 749, 763, 777, 791, 805, 819, 833, 847, 861, 875, 889, 903, 917, 931, 945, 959, 973, 987

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