Mathcad: Count Elements Divisible by 3 Per Row Vector

In Mathcad you can get a per-row vector of counts for elements divisible by 3 either by summing a boolean test across each row or by writing a small programming block. A concise, reliable formula is:
countRow(i) := sum( (M[i,j] mod 3 = 0), j, 0, cols(M)-1 ) then counts := countRow(0..rows(M)-1). For those who prefer imperative code, a short nested-for programming region does the same.

Contents

• Quick vectorized solution (boolean + sum)
• Program module — loop-based (imperative)
• Example and expected output
• Notes: indexing, floating-point and performance
• Sources
• Conclusion

Quick vectorized solution (boolean + sum)

This is the simplest, fastest approach in Mathcad: build a boolean test for divisibility by 3, then sum that test across each row because Mathcad treats true as 1 and false as 0.

Core steps (copy into a Mathcad worksheet):

m := rows(M)
n := cols(M)

countRow(i) := sum( ( M[i, j] mod 3 = 0 ), j, 0, n-1 )

counts := countRow(0..m-1)

• rows(M) and cols(M) give the matrix dimensions.
• sum(expression, j, 0, n-1) sums the expression over column index j.
• The boolean expression (M[i, j] mod 3 = 0) yields 1 for values divisible by 3 and 0 otherwise, so the sum is the count per row.

Why this is recommended: it’s concise, vectorized and generally much faster than explicit nested loops for large matrices. The same boolean + sum trick is discussed in the PTC community examples on counting vector/matrix elements (see the community links in Sources).

Program module — loop-based (imperative)

If you want an explicit programming block (for clarity or because you prefer imperative style), create a Programming Region from Mathcad’s Programming palette and implement nested loops that increment a counter per row.

Readable pseudocode you can paste into a Programming Region (adapt small syntax differences to your Mathcad version as needed):

// Pseudocode for a Mathcad Programming Region
m := rows(M)
n := cols(M)

// Option A: preallocate (preferred) — create a zero vector of length m
// (use your Mathcad version's way to make a zero vector)
counts := zero_vector(m) // replace with the appropriate constructor if needed

for i from 0 to m-1 do
 c := 0
 for j from 0 to n-1 do
 if ( M[i, j] mod 3 = 0 ) then
 c := c + 1
 end if
 end for
 counts[i := c] // assign c to the i-th element of counts (use your version's subscript assignment)
end for

return counts

Notes on implementing the Programming Region:

• Pre-allocating counts (a zero vector of length m) and then assigning counts[i := c] inside the outer loop is the cleanest approach.
• If you prefer to append items, use your version’s concatenation/augment function (e.g., counts := augment(counts, c)) — exact function names vary by Mathcad release.
• This imperative module mirrors the logic shown in the PTC forum examples for counting matrix elements by condition.

Example and expected output

Let M be a 3×3 example matrix:

M :=
[ 3 4 6
 1 9 12
 7 8 10 ]

Using the vectorized solution above:

• Row 0: 3 and 6 are divisible by 3 → count = 2
• Row 1: 9 and 12 are divisible by 3 → count = 2
• Row 2: none are divisible by 3 → count = 0

So counts evaluates to:

counts = [ 2, 2, 0 ]

You can verify the same result with the programming-region implementation.

Notes: indexing, floating-point and performance

• Index base: Mathcad’s default index base is 0 (so ranges 0..n-1 are common). If you use 1-based indices in your sheet, adjust the lower/upper bounds in the sum accordingly (e.g., j, 1, n).
• Non-integers / rounding errors: if matrix entries are floating-point approximations, test divisibility with a tolerance, for example:
• isDivisible := abs(mod(M, 3)) < 1e-10
• or test mod(round(M), 3) = 0 if values should be integers but have tiny numeric noise.
• Negative numbers: mod behavior can depend on version; using round before mod (when appropriate) avoids sign/precision surprises.
• Performance: the boolean+sum method is vectorized and usually fastest in Mathcad Prime and Classic. Use the imperative nested loops only when you need side-effects or very explicit control.
• Version compatibility: the sum(expression, index, a, b) pattern and boolean-as-1 behavior are standard in Mathcad; examples and variations appear in the PTC community threads linked below.

For extra examples and community discussion of both the boolean-sum trick and loop-based approaches see the PTC forum posts in Sources.

Sources

• PTC Mathcad Prime Free Download
• Mathcad Prime Features | PTC
• Count elements in a matrix that meet a condition (PTC Community)
• Counting elements in a vector (PTC Community)
• Counting elements in a matrix (PTC Community)

Conclusion

Use the vectorized boolean + sum pattern in Mathcad to build a vector of per-row counts of elements divisible by 3: define a row function with sum((M[i,j] mod 3 = 0), j, 0, cols(M)-1) and evaluate it over 0..rows(M)-1. It’s concise, fast and easy to adapt; if you need a step-by-step routine, the short nested-for programming module does the same job.