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A divisibility condition can rule out an integer equation or restrict one variable to a remainder class. When the question asks for nonnegative solutions, the remaining inequalities still need to be imposed.

Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 INTEGER RANGE 1102 and tell us your school year and question.

Check a necessary divisibility condition

Every value of 6x + 10y is even when x and y are integers. Thus 6x + 10y = 31 has no integer solutions.

For an original counting problem, instead solve 6x + 10y = 100 with x and y nonnegative integers. Dividing by two gives 3x + 5y = 50.

Use a remainder to parameterise one variable

Reducing the equation modulo three gives 2y ≡ 2, so y ≡ 1 modulo three. Write y = 1 + 3t for an integer t.

Substituting gives 3x = 50 - 5(1 + 3t) = 45 - 15t, hence x = 15 - 5t. The integer family is now explicit, but not every t satisfies the original nonnegative restrictions.

Apply both inequalities

The condition y = 1 + 3t ≥ 0 requires t ≥ 0 for integer t. The condition x = 15 - 5t ≥ 0 requires t ≤ 3.

Therefore t can be 0, 1, 2 or 3. The four pairs are (15,1), (10,4), (5,7) and (0,10). Substitution into 6x + 10y gives one hundred in every case.

Explain why the count is complete

Every solution must satisfy the modulo-three condition, so it must enter this parameter family. The inequalities then include every permitted t, and different t values produce different pairs. This establishes completeness, rather than only finding four examples.

If negative integers were permitted, the same family would have infinitely many members. If strictly positive x were required, the pair with x = 0 would be excluded.

This is an original AMC 12 exercise, not an official question. Remainder conditions, parameterisation and endpoint checks are useful habits for extended number-theory practice, including AIME learning.

Continue reading: After AMC 12: Use a Four-Part Audit to Choose Your AIME Preparation.

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Official information and further reading: Source 1

Dividing an equation by an expression assumes that expression is nonzero. If it can equal zero, solutions may disappear. Factoring can preserve the cases and make the missing condition visible.

Need help choosing a next step? Scan the contact QR code below or contact Hanlin on WeChat. Send AMC12 SINE FACTOR 1101 and tell us your school year and question.

Use the identity without dividing

In this original exercise, solve sin(2x) = sin x for 0 ≤ x < 2π, with angles in radians. Substituting sin(2x) = 2 sin x cos x gives sin x(2 cos x - 1) = 0.

The product is zero when at least one factor is zero. We must therefore examine both sin x = 0 and cos x = 1/2.

Keep the sine-zero case

On the stated interval, sin x = 0 gives x = 0 and x = π. The value 2π would also make sine zero, but it is excluded by the strict upper bound.

At both retained values, sin(2x) and sin x are zero, so they satisfy the original equation. Dividing the original equation by sin x without a separate case would lose these valid solutions.

Solve the other factor on the same interval

The condition cos x = 1/2 gives x = π/3 and x = 5π/3. Together with the sine-zero case, the four solutions are 0, π/3, π and 5π/3.

Substitution checks the second cosine solution as well: sin(10π/3) = sin(4π/3) = -√3/2, which equals sin(5π/3). Do not discard a solution simply because the sine value is negative.

Check cases and endpoints before counting

The factorisation is equivalent to the original equation, and the two cases cover every way the product can vanish. Their listed solutions are distinct, so no duplicate needs removing here.

If division is used instead, explicitly set aside sin x = 0 before dividing on the remaining domain. This is an original AMC 12 learning problem, not an official question. Keeping transformations reversible and recording domain restrictions also supports careful algebra in later AIME practice.

Continue reading: AMC 12 Trigonometry: Square a Sum to Reveal a Double-Angle Value.

Contact Hanlin

Planning a first attempt or reviewing a topic you find difficult? Scan the QR code and send AMC12 SINE FACTOR 1101. Include your school year, country or competition region, experience and the question you want to discuss.

Contact Hanlin on WeChat

On your phone, use the button or save the QR image and scan it within WeChat.

Official information and further reading: Source 1