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Introduction to Number Theory / LEVEL 4 · DIFFICULTY 4/5

Integer Equations and Counts

Combine integer constraints with exact counting.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST NT 4.1Nonnegative Integer SolutionsRead the lesson
  2. MINI QUEST NT 4.2Repeating DecimalsRead the lesson
  3. MINI QUEST NT 4.3Counting Coprime ResiduesRead the lesson
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STAGE NT 4.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Nonnegative Integer Solutions

Useful preparation: Combining Remainder Conditions

Goal: Understand and apply nonnegative integer solutions.

Before you begin: Combining Remainder Conditions

Understand the idea

Integer restrictions turn an equation into a finite count when coefficients and the target are positive. Bounds and parity often eliminate most candidate values before substitution.

x=(N−3y)/2

Choose and carry out a method

For 2x+3y=N, bound y between zero and floor(N/3). Keep only y values making N−3y even, then compute x.

Check the reasoning

Verify x and y are nonnegative integers. Each accepted y determines exactly one x, preventing duplicate counting.

WORKED EXAMPLE 1

How many pairs of nonnegative integers (x,y) satisfy 2x+3y=15?

  1. Use one variable to parameterize the possibilities, then enforce divisibility and nonnegativity.
  2. Try x=0 through 7; retain x only if 15-2x is divisible by 3.
  3. There are 3 pairs. Each retained x determines exactly one y.

3

WORKED EXAMPLE 2

How many pairs of nonnegative integers (x,y) satisfy 2x+3y=18?

  1. Use one variable to parameterize the possibilities, then enforce divisibility and nonnegativity.
  2. Try x=0 through 9; retain x only if 18-2x is divisible by 3.
  3. There are 4 pairs. Each retained x determines exactly one y.

4

Common pitfalls

Possible mix-up: Every y in the numerical range gives a solution.

The remaining numerator must also be divisible by two.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

How does parity let you count the allowed y values without testing all of them?

Preview the eight practice prompts
  1. How many pairs of nonnegative integers (x,y) satisfy 2x+3y=24?
  2. How many pairs of nonnegative integers (x,y) satisfy 2x+3y=27?
  3. How many pairs of nonnegative integers (x,y) satisfy 2x+3y=30?
  4. How many pairs of nonnegative integers (x,y) satisfy 2x+3y=33?
  5. How many pairs of nonnegative integers (x,y) satisfy 2x+3y=36?
  6. How many pairs of nonnegative integers (x,y) satisfy 2x+3y=39?
  7. A store sells packs containing 2 or 3 beads. How many choices of counts of the two pack sizes buy exactly 42 beads? Packs of the same size are indistinguishable. New context
  8. A store sells packs containing 2 or 3 beads. How many choices of counts of the two pack sizes buy exactly 45 beads? Packs of the same size are indistinguishable. New context
Open stage NT 4.1 in the student workspace →

STAGE NT 4.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Repeating Decimals

Useful preparation: Nonnegative Integer Solutions

Goal: Understand and apply repeating decimals.

Before you begin: Nonnegative Integer Solutions

Understand the idea

Multiplying a repeating decimal by a power of ten shifts one complete block. Subtracting the original aligns and removes the infinite repeated tail.

0.overline(ab)=(10a+b)/99

Choose and carry out a method

For a two-digit repeating block, set x equal to the decimal and subtract x from 100x. Solve 99x equal to the block and simplify.

Check the reasoning

A proper decimal below one must give a fraction below one. Leading zeros within a block still count as positions.

WORKED EXAMPLE 1

Write 0.111111… as a fraction, where the two-digit block 11 repeats forever.

  1. Multiply by 100 to align the two-digit repeating tails.
  2. If x=0.1111…, then 100x-x=11, hence 99x=11.
  3. The exact fraction is 1/9; no rounding is needed.

1/9

WORKED EXAMPLE 2

Write 0.121212… as a fraction, where the two-digit block 12 repeats forever.

  1. Multiply by 100 to align the two-digit repeating tails.
  2. If x=0.1212…, then 100x-x=12, hence 99x=12.
  3. The exact fraction is 4/33; no rounding is needed.

4/33

Common pitfalls

Possible mix-up: Divide a repeating two-digit block by 100.

A repeating block uses 99; division by 100 gives a terminating decimal.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

How would the denominator change for a three-digit repeating block?

Preview the eight practice prompts
  1. Write 0.141414… as a fraction, where the two-digit block 14 repeats forever.
  2. Write 0.151515… as a fraction, where the two-digit block 15 repeats forever.
  3. Write 0.161616… as a fraction, where the two-digit block 16 repeats forever.
  4. Write 0.171717… as a fraction, where the two-digit block 17 repeats forever.
  5. Write 0.181818… as a fraction, where the two-digit block 18 repeats forever.
  6. Write 0.191919… as a fraction, where the two-digit block 19 repeats forever.
  7. A display’s exact value repeats the block 20 forever after the decimal point, starting immediately: 0.2020… . Express this reading as a fraction. New context
  8. A display’s exact value repeats the block 21 forever after the decimal point, starting immediately: 0.2121… . Express this reading as a fraction. New context
Open stage NT 4.2 in the student workspace →

STAGE NT 4.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Counting Coprime Residues

Useful preparation: Repeating Decimals

Goal: Understand and apply counting coprime residues.

Before you begin: Repeating Decimals

Understand the idea

An integer is coprime to n exactly when it avoids every prime factor of n. Removing multiples of these primes by inclusion and exclusion counts the permitted residue classes.

φ(n)=n∏(1−1/p) over distinct prime factors p

Choose and carry out a method

For n with prime factors two and three, subtract their multiple counts and restore multiples of six. Express the result as n(1−1/2)(1−1/3).

Check the reasoning

A step size traverses a full circular cycle precisely when it is coprime to the cycle length. Check this on a small circle.

WORKED EXAMPLE 1

How many integers from 1 through 24 are relatively prime to 24?

  1. A step size visits the whole cycle exactly when it is coprime to its length.
  2. Exclude multiples of 2 or 3: φ(24)=24(1-1/2)(1-1/3).
  3. The count is 8; multiples of both primes were excluded twice and must be accounted for once.

8

WORKED EXAMPLE 2

How many integers from 1 through 48 are relatively prime to 48?

  1. A step size visits the whole cycle exactly when it is coprime to its length.
  2. Exclude multiples of 2 or 3: φ(48)=48(1-1/2)(1-1/3).
  3. The count is 16; multiples of both primes were excluded twice and must be accounted for once.

16

Common pitfalls

Possible mix-up: Apply the factor once for every repeated prime.

Use each distinct prime only once.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why does a noncoprime step size revisit its starting point too early?

Preview the eight practice prompts
  1. How many integers from 1 through 192 are relatively prime to 192?
  2. How many integers from 1 through 384 are relatively prime to 384?
  3. How many integers from 1 through 768 are relatively prime to 768?
  4. How many integers from 1 through 1536 are relatively prime to 1536?
  5. How many integers from 1 through 3072 are relatively prime to 3072?
  6. How many integers from 1 through 6144 are relatively prime to 6144?
  7. A 12288-position cycle is traversed using a fixed positive step size at most 12288. How many step sizes visit every position before returning to the start? New context
  8. A 24576-position cycle is traversed using a fixed positive step size at most 24576. How many step sizes visit every position before returning to the start? New context
Open stage NT 4.3 in the student workspace →
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