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How Long Division Works
Long division breaks a complex division into simpler steps: Divide, Multiply, Subtract, Bring down (DMSB). For 125 ÷ 7: 7 goes into 12 once (7), subtract to get 5, bring down 5 to get 55, 7 goes into 55 seven times (49), subtract to get remainder 6. Result: 17 remainder 6.
Quotient and Remainder
In 125 ÷ 7 = 17 R 6, the quotient is 17 (whole number result) and the remainder is 6 (what is left over). This can also be expressed as 125 = 7 × 17 + 6. As a decimal, 125 ÷ 7 = 17.857142... The remainder/divisor (6/7) gives the decimal portion.
Converting Remainders to Decimals
To continue division past the decimal point, add a decimal point and zeros to the dividend, then keep dividing. The remainder 6 from 125÷7 becomes 60, then 4 (since 7×8=56), then 40, then 5 (since 7×5=35), and so on. Some divisions produce repeating decimals.
Division in Programming
In programming, integer division (div or //) discards the remainder: 125 // 7 = 17. The modulo operator (mod or %) returns only the remainder: 125 % 7 = 6. Together, quotient and remainder fully describe the division result. These operations are used in time conversion, pagination, and hashing.
Frequently Asked Questions
Division by zero is undefined because no number multiplied by zero gives a non-zero result. If you think of division as "how many groups of 0 fit into a number," the answer is undefined — you can never reach the dividend by adding zeros.
The modulo (or modulus) operation returns the remainder of a division. 125 mod 7 = 6. It is widely used in programming for cyclic operations, checking divisibility (n mod 2 = 0 means even), and clock arithmetic.
Multiply the quotient by the divisor and add the remainder. The result should equal the dividend: (17 × 7) + 6 = 119 + 6 = 125. This verification is called the division algorithm.
A repeating decimal has a pattern that repeats forever. For example, 1/3 = 0.333... and 1/7 = 0.142857142857... Any fraction with a denominator not solely composed of 2s and 5s produces a repeating decimal.