MATLAB-BOOK The mod function

Subsection B.1 The mod function

The mod function computes the remainder after division. Its syntax is:

`mod` Syntax.

r = mod(a, b)

Returns the remainder r when a is divided by b. The result satisfies 0 <= r < b for positive b.

In other words, mod(a, b) answers the question: “after dividing a by b, what is left over?” For example, if you divide 10 by 3 you get 3 with a remainder of 1, so mod(10, 3) returns 1.

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Two of the most common uses for mod are:

Even/odd test 🔗: mod(n, 2) returns 0 when n is even and 1 when n is odd.
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Divisibility test 🔗: mod(n, k) == 0 is true exactly when n is divisible by k.
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🌌 Example 5. Basic mod Values.

The table below shows several calls to mod and their results.

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Table 6. Sample mod Results

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Expression	Result	Reason
`mod(10, 3)`	`1`	\(10 = 3 \times 3 + 1\)
`mod(12, 4)`	`0`	\(12 = 4 \times 3 + 0\)
`mod(7, 7)`	`0`	any number mod itself is 0
`mod(5, 9)`	`5`	\(5 = 9 \times 0 + 5\)
`mod(0, 6)`	`0`	0 divided by anything leaves remainder 0

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🌌 Example 7. Checking Even and Odd Numbers.

Because mod(n, 2) is 0 for even numbers and 1 for odd numbers, we can use it inside an if-statement to classify a number:

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n = 17;
if mod(n, 2) == 0
    fprintf('%d is even\n', n)
else
    fprintf('%d is odd\n', n)
end

Running this script prints 17 is odd because mod(17, 2) equals 1, not 0.

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🌌 Example 8. Printing Multiples with a Loop.

A for-loop combined with mod is a natural way to select every \(k\)-th value from a range. Here we print all multiples of 3 from 1 to 20:

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for n = 1:20
    if mod(n, 3) == 0
        fprintf('%d\n', n)
    end
end

The output is 3, 6, 9, 12, 15, 18. The condition mod(n, 3) == 0 is true exactly when n is divisible by 3.

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Reading Questions Check Your Understanding

1.

(a) mod(9, 3) equals 0.

mod(9, 3) returns 0.

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True.
Because \(9 = 3 \times 3 + 0\text{,}\) the remainder is 0. Any number that divides evenly leaves a remainder of 0.
False.
Because \(9 = 3 \times 3 + 0\text{,}\) the remainder is 0. Any number that divides evenly leaves a remainder of 0.

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(b) What does mod(7, 4) return?

What does mod(7, 4) return?

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3
\(7 = 4 \times 1 + 3\text{,}\) so the remainder is 3.
1
1 is the quotient (how many times 4 fits into 7), not the remainder.
4
The remainder is always less than the divisor, so it cannot equal 4 here.
0
0 would mean 7 is divisible by 4, but \(7 / 4\) is not a whole number.

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(c) Which expression tests divisibility by 5?

Which expression is true exactly when n is divisible by 5?

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mod(n, 5) == 0
A remainder of 0 means n divides evenly by 5, which is exactly the definition of divisibility.
mod(5, n) == 0
This tests whether 5 is divisible by n, not whether n is divisible by 5.
mod(n, 5) == 5
The remainder from mod(n, 5) is always between 0 and 4, so it can never equal 5.
n / 5 == 0
This is only true when n is 0, not for all multiples of 5 like 5, 10, or 15.

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(d) The result of mod is always less than the divisor.

For a positive divisor b, mod(a, b) always returns a value that is strictly less than b.

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True.
By definition, the remainder when dividing by b satisfies \(0 \leq r < b\text{.}\) Once the remainder would reach b, another full group of b can be subtracted, so the remainder resets to 0.
False.
By definition, the remainder when dividing by b satisfies \(0 \leq r < b\text{.}\) Once the remainder would reach b, another full group of b can be subtracted, so the remainder resets to 0.

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(e) What does mod(0, 6) return?

What does mod(0, 6) return?

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0
Zero divided by any nonzero number gives a quotient of 0 with remainder 0.
6
The remainder is always less than the divisor, so it cannot be 6.
1
Zero is evenly divisible by 6 (zero times), leaving a remainder of 0, not 1.
undefined
mod(0, b) is well defined and equals 0 for any positive b.

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(f) Counting multiples of 4 from 1 to 20.

How many integers from 1 to 20 satisfy mod(n, 4) == 0?

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5
The multiples of 4 in that range are 4, 8, 12, 16, and 20 — five values in all.
4
Don’t forget that 20 is also divisible by 4. The multiples are 4, 8, 12, 16, 20.
6
The next multiple after 20 would be 24, which is outside the range. Count carefully: 4, 8, 12, 16, 20.
3
Three is too few. List them out: \(4, 8, 12, 16, 20\) — that is five values.