Simultaneous remainder compatibility: one shared integer

Simultaneous remainder compatibility asks whether two reports can describe the same integer. In this original paper exercise, a number must leave remainder 2 on division by 6 and remainder 5 on division by 9. The integer 14 satisfies both. Changing the second remainder to 4 makes a shared integer impossible, even though each report can hold separately.

Define both reports for one integer

Let x be an integer. The first report means x - 2 is an integer multiple of 6. The second means x - 5 is an integer multiple of 9. These are exact divisibility statements about the same x. They are stipulated mathematical inputs, without measured rhythms, device readings, real schedules or claims about the pictured catalog items.

Writing x congruent to 2 modulo 6 is another way to state the first report. Writing x congruent to 5 modulo 9 states the second. The modulus belongs to its report: replacing 6 with 9, or attaching the second remainder to the first modulus, would change the input. Keep each remainder and its divisor together when copying the worksheet.

The proposed answer must supply one integer satisfying both reports together. An integer working for the first and a different integer working for the second do not settle that question. For instance, 2 satisfies the first report, while 5 satisfies the second, but neither is a shared witness. We will check the intersection rather than count these separate successes as compatibility.

Find and verify a shared witness

Begin with the first report. Within the integers from 0 through 17, its possibilities are 2, 8 and 14. Their differences from 2 are 0, 6 and 12, all divisible by 6. The next possibility is 20, outside this reporting interval. Choosing this finite interval is a way to display candidates; it has not yet proved the full repeating pattern.

Check those three candidates against the original second report. For 2, the difference from 5 is negative 3, which is not a multiple of 9. For 8, that difference is 3, also not a multiple of 9. For 14, the difference is 9. Thus 14 satisfies the second report as well as the first, giving one actual shared integer.

Retain the two original checks beside it: 14 - 2 = 12 = 2 times 6, and 14 - 5 = 9 = 1 times 9. The same value appears in both calculations. This is stronger than a list of independent possibilities, but it still leaves a question: which other integers satisfy both reports, and how can we prove the answer is complete?

Locate the common-divisor obstruction

Both 6 and 9 are divisible by 3. If an integer x meets both reports, then x - 2 and x - 5 must each be divisible by 3. Subtracting those differences leaves 3, which is divisible by 3. This particular pair passes the necessary check. The requirement concerns the relationship between the two assigned remainders, not an extra restriction on a photographed object.

State the same argument for arbitrary integer reports a and b with the moduli still fixed at 6 and 9. A shared x requires x - a divisible by 6 and x - b divisible by 9. Both differences are then divisible by 3. Their subtraction is b - a, so b - a must be divisible by 3.

That proves necessity. It does not yet show that every pair passing this check has a shared witness. Keep the two directions separate in the explanation. A condition derived from a supposed solution might rule out some inputs without constructing solutions for the rest. For these particular fixed moduli, we can supply an explicit formula that establishes the missing direction.

Construct a witness whenever the check passes

Suppose b - a = 3c for some integer c. Choose x = 4b - 3a. Subtracting a gives x - a = 4b - 4a = 4(b - a) = 12c. This is 6 times the integer 2c. The chosen x therefore satisfies the report with remainder a modulo 6, under exactly the divisibility definition already stated.

Subtracting b from the same x gives x - b = 3b - 3a = 3(b - a) = 9c. That is divisible by 9, so the second report holds too. Since a and b were integers, the constructed x is an integer. Both checks use this one value. We have shown that the difference condition is sufficient as well as necessary.

For the assigned reports a = 2 and b = 5, the formula gives 4 times 5 minus 3 times 2, or 20 - 6 = 14. That agrees with the direct candidate check. The formula was derived for these fixed divisors 6 and 9; it is not a rule to copy unchanged when either divisor is replaced with another number.

Prove the complete shared period

If x and y both satisfy the two reports, their difference d = x - y is divisible by 6 and by 9. Write d = 6u for an integer u. Divisibility by 9 says 6u = 9v for some integer v, so 2u = 3v. Hence 2u is divisible by 3. We need to establish what that implies about u.

An integer u has one of the three remainders 0, 1 or 2 on division by 3. Doubling those possibilities gives remainders 0, 2 or 1, respectively. Only the first makes 2u divisible by 3. Thus u itself is divisible by 3. Write u = 3t; then d = 18t. Every two shared solutions consequently differ by an integer multiple of 18.

Conversely, adding 18t to a shared solution preserves both reports for every integer t. The extra amount is 6 times 3t and also 9 times 2t, so each required difference remains divisible by its own modulus. Combining necessity and sufficiency, the assigned reports have precisely the integer solutions x = 14 + 18t. There are no additional shared classes hidden outside the displayed interval.

The least positive difference between distinct solutions is therefore 18. Every positive difference is 18 times a positive integer, and 18 itself occurs by comparing 14 with 32. This establishes the period rather than guessing it from a short table. Within 0 through 17, exactly one representative appears: 14. The numbers 32 and negative 4 belong to that same repeating class.

Change one report and prove incompatibility

Now preserve a separate version with remainder 2 modulo 6 and remainder 4 modulo 9. If one x satisfied both, x - 2 and x - 4 would each be divisible by 3. Subtracting would make 2 divisible by 3, which is impossible. There is no shared integer, including values outside any finite search interval.

Each changed-version report remains individually possible. The integer 2 satisfies the first; the integer 4 satisfies the second. Those independent witnesses do not satisfy the joint request. Keep this distinction visible when diagnosing the empty intersection. The problem is the two reports' incompatible residues modulo their common divisor, rather than a failure to find a large enough candidate.

Returning to the first reporting interval confirms the contradiction without replacing its proof. The first report still offers 2, 8 and 14. Their remainders on division by 9 are 2, 8 and 5, so none leaves remainder 4. The contradiction above explains why extending this candidate list cannot eventually produce a success for the changed pair.

Keep equivalent descriptions attached to their moduli

A remainder report can be written with a different integer representative of its class. The original first report can use 8 instead of 2 because their difference is 6. The original second can use negative 4 instead of 5 because their difference is negative 9. Divisibility by the respective modulus is unchanged in each rewrite.

With those representatives, the witness formula gives 4 times negative 4 minus 3 times 8, or negative 40. Adding three periods of 18 gives 14. The formula has selected another integer in the same shared class. An intermediate negative value is allowed because the variable was declared to range over all integers; the interval 0 through 17 was only a reporting choice.

More generally, replacing a by a + 6r and b by b + 9s changes 4b - 3a by 36s - 18r. That amount is divisible by 18 for all integers r and s. Equivalent input representatives therefore lead to the same final shared class. Preserve the modulus during such rewrites; adding an arbitrary amount to one report need not preserve its meaning.

Retain exact controls beside the argument

An exact finite check examined all 54 canonical report pairs with a from 0 through 5 and b from 0 through 8. For each pair, it tested every integer from 0 through 17 against both original reports. Exactly 18 pairs were compatible. Each had one representative, equal to 4b - 3a reduced into that interval; each incompatible pair had none.

A further 121 checks rewrote the assigned representatives using r and s from negative five through five. Every resulting witness remained in the class represented by 14. Another 101 checks substituted 14 + 18t for t from negative fifty through fifty into both original reports. These controls agree with the algebra, while the direct witness and difference arguments cover integers outside those tested ranges.

Distinguish the neighboring compatibility questions

The linear congruence solutions guide checks a noninvertible coefficient, reduces one integer relation and lifts its answer back to the original modulus. This exercise intersects two given remainder reports. Its result records one joint witness and a common period, rather than repairing coefficient cancellation or counting lifted classes.

The relative offset compatibility example checks signed exact differences around a closed loop, explicitly without modular wrap. Here one integer must satisfy two divisibility reports. A nonzero difference can preserve a remainder class when it is a multiple of that report's modulus; a zero-total offset loop supplies no substitute proof.

Bound the actual wind-chime accounts

The Orange Cat Bamboo Wind Chime account has two original views showing a sculpted seated cat above attached brown hollow tubes, cords and a central clapper. They depict one assembly from related views. The catalog bamboo label supplies no authenticated material, maker identity, dimensions, delivered quantity or acoustic result.

The Moose silver-tube wind-chime account has five gallery images: an assembly, informational QR image, plaque details and a technical sheet. Its upper moose, cap, tubes, clapper and paired-moose sail form one hanging assembly. Technical material, length, tuning and origin claims remain unverified; the QR image proves no sound result.

Two individually valid reports still need one shared witness.

If you plan to purchase an item on republeq.com, check the particular offer and current checkout before ordering. This guide does not establish that any cryptocurrency payment option is available. If USDT is offered for the intended order, verify the requested asset, network, amount, receiving destination and order reference with the responsible service before authorizing a transfer. Review the actual product identity, selected variant, quantity, delivery details, total charge and applicable terms. Use the confirmed order details when checking the payment instructions. Confirm any fee, conversion term or exchange rate from the current transaction information. Determine the applicable cancellation, return and refund process, including how a refund would be handled for the selected payment method. Keep the relevant instructions and transaction records, with dates and confirmed references. If essential instructions change, ask the responsible service to confirm the change against the exact order. An acknowledgement should identify the order and the agreed details; it does not replace checking whether the selected product is the one you intended to buy. If payment or order status remains uncertain, reconcile that status before considering another transfer. Do not treat an unresolved transaction as evidence that nothing happened. This article supplies no wallet address, transfer guarantee, recovery procedure or assessment of a counterparty. A product enquiry does not authorize a payment or resolve an uncertain order. If support provides an answer, retain the exact question and response so a later reviewer can understand what was confirmed. Decide whether to proceed only when the conditions of the particular transaction are clear, and retain the confirmed transaction information with the exact order.

Save both reports with the intersection

Keep the original pair 2 modulo 6 and 5 modulo 9 beside witness 14 and the complete class 14 + 18t. Save the necessary difference check, explicit construction and period proof together. Retain the changed second remainder 4 as a separate incompatible version. That record shows which two reports were checked and why their individual validity did or did not yield one shared integer.