Which Of The Following Numbers Are Multiples Of 7

Which of the following numbers are multiples of 7?
Understanding how to spot a multiple of 7 is a fundamental skill in arithmetic that builds a strong foundation for more advanced math topics such as fractions, algebra, and number theory. This article explains what a multiple of 7 is, shows reliable methods to test divisibility, walks through numerous examples, and offers practice problems to reinforce the concept. By the end, you’ll be able to quickly determine whether any given integer belongs to the set of multiples of 7.

Introduction to Multiples of 7

A multiple of 7 is any integer that can be expressed as (7 \times n), where (n) is a whole number (including zero). In other words, when you divide the number by 7, the remainder is zero. The sequence begins with 0, 7, 14, 21, 28, … and continues infinitely in both the positive and negative directions. Recognizing these numbers quickly is useful in everyday calculations, problem‑solving, and even in programming loops that rely on regular intervals.

How to Identify a Multiple of 7

The Direct Division Method The most straightforward way is to perform the division:

[ \text{If } \frac{N}{7} \text{ yields an integer, then } N \text{ is a multiple of 7.} ]

For example, (56 \div 7 = 8) with no remainder, so 56 is a multiple of 7. Conversely, (58 \div 7 = 8) remainder 2, so 58 is not a multiple of 7.

The Divisibility Rule for 7

While direct division works, a handy mental shortcut can speed up the process, especially for larger numbers. The rule is:

Take the last digit, double it.
Subtract that result from the rest of the number (the number formed by removing the last digit).
If the difference is 0 or a multiple of 7, then the original number is a multiple of 7. 4. Repeat the steps if the resulting number is still large.

Example: Test 203.

Last digit = 3 → double = 6.
Rest of number = 20.
Subtract: (20 - 6 = 14).
14 is a multiple of 7, so 203 is also a multiple of 7.

Another example: Test 1,254.

Last digit = 4 → double = 8.
Rest = 125 → (125 - 8 = 117). - Repeat: last digit 7 → double 14; rest 11 → (11 - 14 = -3).
(-3) is not a multiple of 7, so 1,254 is not a multiple of 7.

This rule leverages the fact that (10 \equiv 3 \pmod{7}), allowing a quick reduction without a calculator.

Step‑by‑Step Procedure (Algorithm)

If you prefer a checklist, follow these steps:

Write down the number you want to test.
Isolate the units digit (the rightmost digit).
Double that digit.
Subtract the doubled value from the number formed by the remaining digits.
Check the result:
- If it is 0, 7, -7, 14, -14, … → the original number is a multiple of 7.
- If not, repeat the process with the new number. 6. Stop when the number is small enough to recognize instantly (usually under 20).

Examples: Identifying Multiples of 7 from a List

Consider the following set of numbers:

[ { 0, 7, 13, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133 } ]

Applying the divisibility rule (or simple division) shows that all numbers in this list are multiples of 7 except 13. Here’s a quick verification:

Number	Last digit ×2	Rest – (last×2)	Result	Multiple of 7?
0	0×2 = 0	0 – 0 = 0	0	✅
7	7×2 = 14	0 – 14 = -14	-14	✅
13	3×2 = 6	1 – 6 = -5	-5	❌
21	1×2 = 2	2 – 2 = 0	0	✅
28	8×2 = 16	2 – 16 = -14	-14	✅
35	5×2 = 10	3 – 10 = -7	-7	✅
42	2×2 = 4	4 – 4 = 0	0	✅
49	9×2 = 18	4 – 18 = -14	-14	✅
56	6×2 = 12	5 – 12 = -7	-7	✅
63	3×2 = 6	6 – 6 = 0	0	✅
70	0×2 = 0	7 – 0 = 7	7	✅
77	7×2 = 14	7 – 14 = -7	-7	✅
84	4×2 = 8	8 – 8 = 0	0	✅
91	1×2 = 2	9 –

Completing the illustration

Continuing the quick check for the remaining entries:

Number	Last digit × 2	Rest – (last × 2)	Result	Multiple of 7?
91	1 × 2 = 2	9 − 2 = 7	7	✅
98	8 × 2 = 16	9 − 16 = ‑7	‑7	✅
105	5 × 2 = 10	10 − 10 = 0	0	✅
112	2 × 2 = 4	11 − 4 = 7	7	✅
119	9 × 2 = 18	11 − 18 = ‑7	‑7	✅
126	6 × 2 = 12	12 − 12 = 0	0	✅
133	3 × 2 = 6	13 − 6 = 7	7	✅

All numbers in the set are therefore multiples of 7 except for 13, which fails the test at the first iteration (‑5 is not a multiple of 7).

Extending the technique to larger figures

When the intermediate result still exceeds a comfortable mental threshold, the same subtraction‑and‑double‑digit step can be applied repeatedly. For instance, testing 987 654:

Units digit = 4 → double = 8; remaining part = 98 765 → 98 765 − 8 = 98 757.
Units digit = 7 → double = 14; remaining part = 9 875 → 9 875 − 14 = 9 861.
Units digit = 1 → double = 2; remaining part = 986 → 986 − 2 = 984.
Units digit = 4 → double = 8; remaining part = 98 → 98 − 8 = 90. 5. 90 is not a multiple of 7, so 987 654 is not divisible by 7.

The process can be halted as soon as the reduced number is small enough to recognise instantly (e.g., 0, ±7, ±14, ±21).

A quick sanity check

Because the rule is based on the congruence (10 \equiv 3 \pmod{7}), each iteration effectively replaces the original number with an equivalent residue class modulo 7. Consequently, the algorithm never introduces false positives: if the final remainder is 0 (or a multiple of 7), the original integer shares that remainder, and if it is any other value, divisibility fails.

ConclusionThe “double‑the‑last‑digit and subtract” method provides a swift, calculator‑free pathway to decide whether a number is a multiple of 7. By isolating the units digit, doubling it, and subtracting from the truncated prefix, we shrink the problem size while preserving the modular relationship. Repeating the operation until the intermediate result is trivially recognizable yields a definitive answer. This technique is especially handy for mental arithmetic, classroom drills, or any situation where rapid verification of divisibility by 7 is required.