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Introduction to Number Theory / LEVEL 2 · DIFFICULTY 2/5

Factorials and Representations

Understand place value and repeated factors.

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

Choose an island to read its lesson.

  1. MINI QUEST NT 2.1Prime Powers in FactorialsRead the lesson
  2. MINI QUEST NT 2.2Place Value in Other BasesRead the lesson
  3. MINI QUEST NT 2.3Computing with RemaindersRead the lesson
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STAGE NT 2.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Prime Powers in Factorials

Useful preparation: Summing Divisors

Goal: Understand and apply prime powers in factorials.

Before you begin: Summing Divisors

Understand the idea

A factorial contains one factor p from every multiple of p, an additional one from every multiple of p², and so on. Counting these layers captures numbers with repeated prime factors.

vₚ(n!)=⌊n/p⌋+⌊n/p²⌋+…

Choose and carry out a method

Add floor(n/p), floor(n/p²), and subsequent terms until they become zero. For trailing decimal zeros, compare the exponents of two and five.

Check the reasoning

A multiple of p² must contribute at least twice. Counting only multiples of p misses this extra contribution.

WORKED EXAMPLE 1

What is the exponent of 5 in the prime factorization of 17!?

  1. Count a contribution for each multiple of 5, another for each multiple of 25, and so on.
  2. Add floor(17/5)+floor(17/25)+floor(17/125)+… .
  3. The total is 3. For trailing zeros, factors of 2 are more plentiful, so factors of 5 determine the count.

3

WORKED EXAMPLE 2

What is the exponent of 5 in the prime factorization of 22!?

  1. Count a contribution for each multiple of 5, another for each multiple of 25, and so on.
  2. Add floor(22/5)+floor(22/25)+floor(22/125)+… .
  3. The total is 4. For trailing zeros, factors of 2 are more plentiful, so factors of 5 determine the count.

4

Common pitfalls

Possible mix-up: Every multiple of p contributes exactly once.

Multiples of higher powers contribute additional factors.

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 do factors of five control the trailing zeros of a factorial?

Preview the eight practice prompts
  1. What is the exponent of 5 in the prime factorization of 32!?
  2. What is the exponent of 5 in the prime factorization of 37!?
  3. What is the exponent of 5 in the prime factorization of 42!?
  4. What is the exponent of 5 in the prime factorization of 47!?
  5. What is the exponent of 5 in the prime factorization of 52!?
  6. What is the exponent of 5 in the prime factorization of 57!?
  7. The product 1·2·…·62 is written in decimal notation. How many trailing zeros does it have? New context
  8. The product 1·2·…·67 is written in decimal notation. How many trailing zeros does it have? New context
Open stage NT 2.1 in the student workspace →

STAGE NT 2.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Place Value in Other Bases

Useful preparation: Prime Powers in Factorials

Goal: Understand and apply place value in other bases.

Before you begin: Prime Powers in Factorials

Understand the idea

A positional numeral represents a sum of digits times powers of its base. The positions count groups of b, then groups of b groups, rather than always groups of ten.

(d₂d₁d₀)ᵦ=d₂b²+d₁b+d₀

Choose and carry out a method

Starting from the right, label weights 1, b, b² and so on. Multiply each digit by its weight and add to convert to decimal.

Check the reasoning

Every digit must be smaller than the base. Conversion changes the written representation, not the represented integer.

WORKED EXAMPLE 1

Convert the base-6 numeral 123 to base ten.

  1. Place values in base b are powers of b.
  2. 123 in base 6 means 1·6²+2·6+3.
  3. The decimal value is 51; every displayed digit is smaller than the base.

51

WORKED EXAMPLE 2

Convert the base-7 numeral 123 to base ten.

  1. Place values in base b are powers of b.
  2. 123 in base 7 means 1·7²+2·7+3.
  3. The decimal value is 66; every displayed digit is smaller than the base.

66

Common pitfalls

Possible mix-up: Read a base-b numeral as a decimal numeral.

Its place weights are powers of b.

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

Convert your result back by repeated division by the base.

Preview the eight practice prompts
  1. Convert the base-9 numeral 123 to base ten.
  2. Convert the base-10 numeral 123 to base ten.
  3. Convert the base-11 numeral 123 to base ten.
  4. Convert the base-12 numeral 123 to base ten.
  5. Convert the base-13 numeral 123 to base ten.
  6. Convert the base-14 numeral 123 to base ten.
  7. A counter uses base 15. Its three displayed digits are 1, 2, and 3. What ordinary base-ten count do they represent? New context
  8. A counter uses base 16. Its three displayed digits are 1, 2, and 3. What ordinary base-ten count do they represent? New context
Open stage NT 2.2 in the student workspace →

STAGE NT 2.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Computing with Remainders

Useful preparation: Place Value in Other Bases

Goal: Understand and apply computing with remainders.

Before you begin: Place Value in Other Bases

Understand the idea

Congruent integers differ by a multiple of the modulus. Addition and multiplication preserve this relationship, allowing large expressions to be reduced before arithmetic grows unwieldy.

a≡r, b≡s (mod m) ⇒ ab+a≡rs+r

Choose and carry out a method

Reduce each input to a remainder, carry out the addition and multiplication, then reduce the final result again.

Check the reasoning

The standard remainder is at least zero and smaller than the positive modulus. Division needs extra conditions and is not automatically valid.

WORKED EXAMPLE 1

Find the least nonnegative remainder of 18·14+18 when divided by 4.

  1. Reduce factors before multiplying, then reduce the final sum.
  2. 18≡2 and 14≡2 modulo 4, so the expression is congruent to 2·2+2.
  3. The least nonnegative representative is 2, which lies between 0 and 3.

2

WORKED EXAMPLE 2

Find the least nonnegative remainder of 23·18+23 when divided by 5.

  1. Reduce factors before multiplying, then reduce the final sum.
  2. 23≡3 and 18≡3 modulo 5, so the expression is congruent to 3·3+3.
  3. The least nonnegative representative is 2, which lies between 0 and 4.

2

Common pitfalls

Possible mix-up: A remainder may equal the modulus.

Replace it by zero; standard remainders stop at m−1.

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

Explain why replacing an input by a congruent one leaves the result unchanged.

Preview the eight practice prompts
  1. Find the least nonnegative remainder of 30·22+30 when divided by 7.
  2. Find the least nonnegative remainder of 35·26+35 when divided by 8.
  3. Find the least nonnegative remainder of 37·30+37 when divided by 9.
  4. Find the least nonnegative remainder of 42·34+42 when divided by 10.
  5. Find the least nonnegative remainder of 47·34+47 when divided by 11.
  6. Find the least nonnegative remainder of 49·38+49 when divided by 12.
  7. A dial has 13 positions numbered from 0. Starting at 0, a machine advances 54 steps on each of 42 runs, then advances 54 more steps. At which position does it stop? New context
  8. A dial has 14 positions numbered from 0. Starting at 0, a machine advances 59 steps on each of 46 runs, then advances 59 more steps. At which position does it stop? New context
Open stage NT 2.3 in the student workspace →
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