# GRE Divisibility and Remainders: Worked Problems | topin

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## Translate every remainder into an integer equation

[ETS](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html) names divisibility, factorisation, prime numbers and remainders in GRE Quant’s arithmetic scope. For a positive divisor d, the remainder r in integer division satisfies 0 ≤ r < d, and n = dq + r for an integer quotient q. The remainder is a whole-number leftover, not the decimal part of n/d. Write that form as soon as the condition appears.

For example, if n leaves remainder four when divided by seven, write n = 7q + 4 with q an integer. Candidate values differ by seven. If n is positive, the admissible q values must also make the whole expression positive; the quotient is not automatically positive in every problem. A condition on n cannot be silently transferred to a different variable.

Check the domain first. Under ETS’s conventions, “x is a number” allows fractions and negatives, so remainder reasoning only applies once the question says the values are integers. That matters most in [Quantitative Comparison](/gre/quant/quantitative-comparison). If n is a positive integer that leaves remainder 5 when divided by 6, compare Quantity A, the remainder when n is divided by 3, with Quantity B, 2\. Writing n = 6q + 5 = 3(2q + 1) + 2 shows that A is always 2, so the answer is C, without testing a single value. The [GRE Quant section guide](/gre/quant) sets out all five question formats.

## Choose factors, a remainder form or a short list

Use prime factorisation when a question asks whether one integer divides another, or asks for a greatest common factor or least common multiple. Use n = dq + r when a remainder is stated. Use a short candidate list when there is a tight interval. These routes can combine, but one clear representation is usually better than beginning with several unrelated tests.

Our decision table is a recommended work order. It is not an ETS formula-frequency list. For greatest common factors, retain the shared prime factors with the smaller exponents. For least common multiples of positive integers, retain every required prime factor with the larger exponents. These rules follow directly from the factors needed to divide each original number.

__Match the integer condition to the representation__
| Question feature                       | First move                                    | Check                          |
| -------------------------------------- | --------------------------------------------- | ------------------------------ |
| Remainder r on division by d           | Write n = dq + r                              | 0 ≤ r < d                      |
| Common divisors                        | Prime-factorise and compare smaller exponents | Result divides both numbers    |
| Common multiples                       | Use larger required prime exponents           | Both numbers divide the result |
| Two conditions inside a short interval | List from the larger divisor                  | Test the second condition      |
| Large power remainder                  | Look for a repeating remainder cycle          | Match the exponent position    |

## Original problem: intersect remainder conditions

Original question: n is an integer with 20 < n < 70\. Dividing n by six leaves remainder five, and dividing n by four leaves remainder one. Find all possible values of n. Start with the six-divisor condition: the candidates in the interval are 23, 29, 35, 41, 47, 53, 59 and 65\. Test these against division by four.

The values 29, 41, 53 and 65 leave remainder one. Their difference is 12, which is the least common multiple of six and four. The pattern can be written n = 12k + 5, restricted by the interval. Check 29 = 6 × 4 + 5 = 4 × 7 + 1\. A candidate must satisfy both conditions; meeting the first alone is insufficient.

Notice that the product 6 × 4 = 24 is not the smallest step in this example because the divisors share a factor. A formula based on their product can skip valid values. When the interval is short, direct listing avoids unnecessary machinery. If the pattern is used, first verify a valid starting value and the common step, then enforce strict or inclusive endpoints exactly.

## Original problem: carry remainders through an expression

Original question: n leaves remainder three when divided by five. What remainder does 2n + 4 leave when divided by five? Write n = 5q + 3\. Then 2n + 4 = 10q + 10 = 5(2q + 2), so the remainder is zero. Multiplying the stated remainder by two and adding four gives ten, but ten itself is not an allowable remainder for divisor five.

A second original question asks for the remainder of n^2 on division by five under the same condition. Expanding gives (5q + 3)^2 = 25q^2 + 30q + 9\. The first two terms are multiples of five, and nine leaves remainder four. You can therefore use the smaller representative three, square it, and reduce the result to an allowed remainder.

Be cautious with division. From n leaving remainder three modulo five, you cannot simply divide that remainder by two and call 1.5 the remainder of n/2\. The new expression may not even be an integer. Addition and multiplication preserve the integer-multiple structure; division introduces extra conditions. Write out the original equation if a supposed shortcut creates a fraction where an integer remainder is required.

## Original problem: reduce a large exponent using a cycle

Original question: what is the remainder when 3^47 is divided by seven? The remainders of successive powers are 3, 2, 6, 4, 5 and 1\. Multiplying the final remainder by three returns to three, so this cycle repeats every six exponents. Since 47 = 6 × 7 + 5, use the fifth position of the cycle, giving remainder five.

The cycle starts at exponent one. An exponent divisible by six uses the sixth position, remainder one, rather than an invented zeroth list entry. Check a small exponent such as six or seven when setting up the index. A calculator display of the huge power is unnecessary and may not preserve a useful exact integer; the small remainders carry all the relevant information.

The cycle method also works for units digits by considering division by ten. For 7^23, the units-digit cycle is 7, 9, 3, 1, with length four. The exponent 23 leaves position three, so the units digit is three. The divisor and base decide the cycle; do not import the cycle length from a different example without calculating it.

## Original problem: test divisibility through prime factors

Original question: let N = 2^3 × 3^2 × 5\. Is N divisible by 72, 90 and 120? Factor each candidate. Since 72 = 2^3 × 3^2, every required prime exponent is present. Since 90 = 2 × 3^2 × 5, it also divides N. Since 120 = 2^3 × 3 × 5, it divides N too. You do not need to expand N to 360.

By contrast, 144 = 2^4 × 3^2 does not divide N because it requires four factors of two and N has only three. A divisibility statement concerns all required prime factors, including their multiplicities. Knowing that N is even or divisible by three is too weak to establish divisibility by a larger composite integer. Mark the missing exponent rather than testing a rounded decimal quotient.

A related original question asks for the greatest power of five dividing 18!. Count factors of five among its factors: 5, 10 and 15 each contribute one, and there is no multiple of 25\. The exponent is three. For larger factorials, multiples of 25 contribute an extra five, multiples of 125 another, and so on. This is a factor-counting deduction, not a claim that factorial questions have a fixed test frequency.

## Check domains, endpoints and the allowed remainder

Our review checklist starts with three constraints: integer domain, divisor sign and remainder bounds. Then check the result in the original statement. A remainder equal to the divisor must reduce to zero; a negative residual must be rewritten with the nonnegative remainder for a positive divisor. For example, −1 = 5(−1) + 4, so its remainder on division by five is four.

Practise one restricted interval, one transformed expression, one power cycle and one prime-factor comparison. Vary whether interval endpoints are included. The [inequalities guide](/articles/gre-inequalities-absolute-values) develops those boundary decisions. After the concepts are stable, use [topin’s free GRE mock](/gre/practice-test), marked on the official scale, to check whether the integer constraints remain visible under mixed-topic timing.

For [select-one-or-more questions](/gre/quant/multiple-choice), test every option against every condition, even after finding one valid value. Record whether a miss came from incomplete checking, an incorrect cycle index or a lost integer restriction. The next practice set should target that cause. Recomputing the same expression more quickly does not repair a model that allowed forbidden numbers.

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## FAQs

What is the equation for an integer remainder?

For positive divisor d, write n = dq + r with integer q and 0 ≤ r < d. This records the quotient and the allowed whole-number leftover.

Can a remainder equal the divisor?

No. If the leftover reaches the divisor, one more full divisor can be removed. For positive divisor d, valid remainders are integers from zero through d − 1.

Why use the least common multiple for two remainder conditions?

A common step must preserve both division conditions. The least common multiple is the smallest positive step divisible by both divisors. You still need a starting value that satisfies both conditions.

How do I find the remainder of a large power?

Compute successive small remainders until a cycle is established, then locate the exponent within that cycle. Check the indexing, especially when the exponent is a multiple of the cycle length.

Is one a prime number on the GRE?

No. ETS’s conventions specify that prime numbers are greater than one. A prime positive integer has exactly two positive divisors: one and itself. Keep one separate when reasoning about prime factors.

## Sources (checked 5 October 2026)

* [ETS: Quantitative Reasoning overview and calculator guidance (checked 5 October 2026)](https://www.ets.org/gre/test-takers/general-test/prepare/content/quantitative-reasoning.html)
* [ETS: GRE Math Review (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-review.pdf)
* [ETS: GRE Mathematical Conventions (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/gre-math-conventions.pdf)
* [ETS: Guidelines specific to the on-screen calculator (checked 5 October 2026)](https://www.ets.org/content/dam/ets-org/pdfs/gre/on-screen-calculator-guidelines.pdf)

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