Modulo Order Of Operations Calculator - PEMDAS Step by Step

The modulo order of operations calculator is a math tool that takes three integers a, b, and c plus the two operators between them, and it returns the result of the expression a op1 b op2 c using the PEMDAS precedence rule that the modulo operation shares with multiplication and division. The page shows which step the calculator runs first, which step it runs second, and what the final integer is for any three-operand chain that mixes modulo with the four basic operators.

• • Homework check on PEMDAS with mod: Confirm a textbook answer for an expression like 10 + 17 mod 5 by reading the step-by-step breakdown.
• • Classroom demonstration: Project the steps on a board to show that mod is in the same precedence tier as multiplication and division.
• • Coding bug hunting: Trace the precedence of a Python, JavaScript, or C expression that mixes % or mod with / and +.
• • Mental math sanity check: Plug a tricky chain into the page to confirm whether the answer is the mod result first or the additive combination second.

The rule the page enforces is the rule taught in school: multiplication, division, and modulo all sit on the same precedence tier and collapse left to right; addition and subtraction sit on the lower tier and also collapse left to right. When one operator is in the multiplicative tier and the other is in the additive tier, the multiplicative step always runs first, so 10 + 17 mod 5 evaluates as 10 + (17 mod 5) = 12, not (10 + 17) mod 5 = 2.

For a flat a mod n result with no precedence chain, Modulo Calculator returns the remainder directly so the user does not have to build a three-operand expression when the question is just what the remainder of a single mod is.

The calculator takes the three operands and the two operators, builds a flat token list, and walks two precedence passes. The first pass collapses *, /, and mod in left-to-right order; the second pass collapses + and - in left-to-right order. Each collapse records one step, and the number that remains after the second pass is the result. Division uses floored division and modulo uses floored modulo so the answer matches what most math textbooks and programming languages teach.

• a, b, c: The three integer operands you type. They can be positive or negative in the calculator range, but c must be non-zero if op2 is / or mod, and b must be non-zero if op1 is / or mod.
• op1, op2: The two operators, each chosen from +, -, *, /, mod. The mod operator is treated as a multiplicative-tier operator, so it runs in pass 1 alongside * and /.
• pass 1: The multiplicative pass. Collapses any pair joined by *, /, or mod in left-to-right order. After pass 1, no multiplicative operator is left in the token list.
• pass 2: The additive pass. Collapses any pair joined by + or - in left-to-right order. After pass 2, exactly one number remains, and that number is the result.
• result: The integer left in the token list after both passes. Returned as null if any pass tried to divide by zero.

The result panel reports the final integer, the flattened expression the calculator actually evaluated, and the recorded steps in the order they happened.

According to Wikipedia, Order of operations, multiplication, division, and modulo share a precedence tier and collapse left to right

When the question shifts from precedence to grouping, Is Modulo Associative Calculator tests the (a mod b) mod c versus a mod (b mod c) ordering on the same mod operator this page uses, so the same-page user can see why the left-to-right fold on the multiplicative tier matters.

These four small ideas show up every time the modulo order of operations is taught. Knowing the language keeps the steps short and specific.

These four ideas are the building blocks of every precedence decision. PEMDAS names the tiers, the precedence tier is the group the mod step belongs to, the left-to-right fold is what happens inside a tier, and the floored remainder is the convention for the mod step when the operands are negative.

For a wider survey of how multiple operators sit next to each other and collapse step by step, Binary Operations Calculator walks through add, subtract, multiply, divide, AND, OR, XOR, and NOT on binary operands, which is the same multi-operator walkthrough this page runs on the PEMDAS tiers with mod in the mix.

Pick the three operands and the two operators, and the page runs the two precedence passes for you. The default chain 10 + 17 mod 5 already produces the standard mod-first example.

1. 1 Enter the first number: Type a value for a. The default is 10, but any integer in the calculator range is allowed, positive or negative.
2. 2 Pick the first operator: Choose op1 from +, -, x, /, or mod. The default is +. Modulo runs in pass 1.
3. 3 Enter the middle number: Type a value for b. The default is 17. If op1 is / or mod, b must be non-zero or the page will switch the result to Undefined.
4. 4 Pick the second operator: Choose op2 from +, -, x, /, or mod. The default is mod.
5. 5 Enter the last number: Type a value for c. The default is 5. If op2 is / or mod, c must be non-zero.
6. 6 Read the result and the steps: Look at the primary result, the flattened expression, and the step-by-step breakdown.

Try the standard mod-first example: leave a = 10, op1 = +, b = 17, op2 = mod, c = 5. The page reports result = 12 with steps 'Step 1: 17 mod 5 = 2. Step 2: 10 + 2 = 12.' Now switch op1 to - and op2 to + to get 10 - 17 + 5. The result is -2 with steps 'Step 1: 10 - 17 = -7. Step 2: -7 + 5 = -2.', showing the additive left-to-right fold.

When the mod step has to handle a negative dividend or a negative divisor, Modulo Of Negative Numbers Calculator reports the floored, truncated, and Euclidean remainders so the mod step on this page matches the convention the user has chosen for the rest of the chain.

Running the precedence rule through a small page buys speed, a clean step-by-step walkthrough, and a way to talk about mod-first vs add-first with concrete numbers.

The result is the headline output, but the steps list is what makes the page useful for showing work. The user can read which operator the page treated as the first step and which as the second step, so the precedence decision is visible at a glance.

For a survey of the practical situations where the mod step on this page actually shows up, Uses Of Modulo Calculator walks through clock arithmetic, hashing, parity checks, and the rest of the standard mod use cases so the user can see where the precedence rule matters in real work.

A handful of inputs and choices decide which operator the page treats as the first step. Knowing them keeps the precedence reading honest.

• • The page is restricted to three operands and two operators, so it cannot show precedence for longer chains like a + b * c / d - e.
• • The page uses floored modulo, not the truncated or Euclidean convention, so the mod step may differ from C, Java, or JavaScript.

According to Wikipedia, Modulo operation, modulo by zero is undefined, and the floored convention keeps the result in the same sign range as the divisor

According to Wikipedia, Operator-precedence parser, an operator-precedence parser collapses the highest-precedence operators first and falls through to lower tiers

When the additive or multiplicative tier is the focus, Associative Property Calculator tests the associative property on the four basic operations, which is the same fold this page runs in pass 2 and pass 1, so the precedence rule and the grouping rule stay consistent across the two pages.