How Many 3 Digit Numbers Are Divisible by 7?

Understanding how many 3-digit numbers are divisible by 7 is a classic math problem often explored in competitive exams and school-level number theory. This article explains the logic behind solving it step by step, while also covering related concepts like divisibility, arithmetic sequences, and key terms for better comprehension.

What Are 3-Digit Numbers?

A 3-digit number is any number from 100 to 999. These are numbers with exactly three digits and represent the smallest to largest values in that numerical range.

What Does Divisible by 7 Mean?

If a number is divisible by 7, it means that when you divide it by 7, the remainder is 0. In other words:

N % 7 = 0Where % denotes the remainder.

For example:

105 ÷ 7 = 15 → ✔️ divisible

106 ÷ 7 = 15.14 → ❌ not divisible

How to Find How Many 3-Digit Numbers Are Divisible by 7

We’ll use a simple method involving Arithmetic Progression (AP).

Step 1: Find the first 3-digit number divisible by 7

We start at 100 and go upward:

100 ÷ 7 ≈ 14.28 → next whole number = 15

15 × 7 = 105

So, the first 3-digit number divisible by 7 is 105.

Step 2: Find the last 3-digit number divisible by 7

We stop at 999 and go downward:

999 ÷ 7 ≈ 142.71 → take floor value = 142

142 × 7 = 994

So, the last 3-digit number divisible by 7 is 994.

Step 3: Use the AP Formula

Now we form an AP: 105, 112, 119, ..., 994

First term (a) = 105

Common difference (d) = 7

Last term (l) = 994

We use the AP formula:

n = ((l - a) / d) + 1

n = ((994 - 105) / 7) + 1

n = (889 / 7) + 1 = 127 + 1 = 128

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

List of Last 3-Digit Number Divisible by 7

Here is the complete list of all 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

These numbers follow an arithmetic progression with a common difference of 7.

Quick Tips for Divisibility by 7

• There’s no simple “last digit rule” like 5 or 2.
• Divide the number directly and check the remainder.
• For quick estimates, memorize the first few multiples of 7: 7, 14, 21, ..., 98.

How many 3 digit numbers are divisible by 7 Worksheet

Q. What is the smallest 3-digit number?

Ans. 100

Q. What is the largest 3-digit number?

Ans. 999

Q. Define what it means for a number to be divisible by 7.

Ans. A number is divisible by 7 if it gives a remainder of 0 when divided by 7 (i.e., N mod  7=0)

Q. What is the formula for finding the total terms in an arithmetic progression (AP)?

Ans. n= (l−a)/ d +1

Q. First 3-digit number divisible by 7

Ans. 105 (Because 7×15=105)

Q. Last 3-digit number divisible by 7

Ans. 994 (Because 7×142=994)

Q. First five terms in AP

Ans. 105, 112, 119, 126, 133

Q. Use AP formula to find total terms

Ans. 𝑛= 994−105 / 7 + 1

= 889/7 + 1

= 127 + 1

= 128

Q. 203 divisible by 7? Show your work.

Ans. 203÷7=29

Yes, 203 is divisible by 7.

Q. List all 3-digit numbers divisible by 7 between 200 and 300

Ans. 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294

Q. What is the 15th number in the sequence?

Ans. Use: a+(n−1)d

= 105+14×7

= 105+98

=203