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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.

MINI QUEST NT 2.1Prime Powers in FactorialsRead the lesson
MINI QUEST NT 2.2Place Value in Other BasesRead the lesson
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 $\displaystyle p^{2}$ , and so on. Counting these layers captures numbers with repeated prime factors.

\displaystyle v_{p}\, \left(n!\right)

=⌊

\displaystyle \frac{n}{p}

⌋+⌊

\displaystyle \frac{n}{p^{2}}

⌋+…

Choose and carry out a method

Add floor( $\displaystyle \frac{n}{p}$ ), floor( $\displaystyle \frac{n}{p^{2}}$ ), and subsequent terms until they become zero. For trailing decimal zeros, compare the exponents of two and five.

Check the reasoning

A multiple of $\displaystyle p^{2}$ 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 $\displaystyle 17!$ ?

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

WORKED EXAMPLE 2

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

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

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

Problem 1
What is the exponent of 5 in the prime factorization of $\displaystyle 32!$ ?
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Problem 2
What is the exponent of 5 in the prime factorization of $\displaystyle 37!$ ?
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Problem 3
What is the exponent of 5 in the prime factorization of $\displaystyle 42!$ ?
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Problem 4
What is the exponent of 5 in the prime factorization of $\displaystyle 47!$ ?
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Problem 5
What is the exponent of 5 in the prime factorization of $\displaystyle 52!$ ?
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Problem 6
What is the exponent of 5 in the prime factorization of $\displaystyle 57!$ ?
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Problem 7New context
The product $\displaystyle 1 \cdot 2$ ·…·62 is written in decimal notation. How many trailing zeros does it have?
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Problem 8New context
The product $\displaystyle 1 \cdot 2$ ·…·67 is written in decimal notation. How many trailing zeros does it have?
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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.

\displaystyle \left(d_{2}\, d_{1}\, d_{0}\right)

ᵦ=

\displaystyle d_{2}\, b^{2} + d_{1}\, b + d_{0}

Choose and carry out a method

Starting from the right, label weights 1, b, $\displaystyle b^{2}$ 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.

Place values in base b are powers of b.
123 in base 6 means $\displaystyle 1 \cdot 6^{2} + 2 \cdot 6 + 3$ .
The decimal value is 51; every displayed digit is smaller than the base.

WORKED EXAMPLE 2

Convert the base-7 numeral 123 to base ten.

Place values in base b are powers of b.
123 in base 7 means $\displaystyle 1 \cdot 7^{2} + 2 \cdot 7 + 3$ .
The decimal value is 66; every displayed digit is smaller than the base.

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

Problem 1
Convert the base-9 numeral 123 to base ten.
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Problem 2
Convert the base-10 numeral 123 to base ten.
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Problem 3
Convert the base-11 numeral 123 to base ten.
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Problem 4
Convert the base-12 numeral 123 to base ten.
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Problem 5
Convert the base-13 numeral 123 to base ten.
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Problem 6
Convert the base-14 numeral 123 to base ten.
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Problem 7New context
A counter uses base 15. Its three displayed digits are 1, 2, and 3. What ordinary base-ten count do they represent?
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Problem 8New context
A counter uses base 16. Its three displayed digits are 1, 2, and 3. What ordinary base-ten count do they represent?
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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.

\displaystyle a \equiv r

\displaystyle b \equiv s

(mod m) ⇒

\displaystyle ab + a \equiv 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 $\displaystyle 18 \cdot 14 + 18$ when divided by 4.

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

WORKED EXAMPLE 2

Find the least nonnegative remainder of $\displaystyle 23 \cdot 18 + 23$ when divided by 5.

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

Common pitfalls

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

Replace it by zero; standard remainders stop at $\displaystyle m - 1$ .

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

Problem 1
Find the least nonnegative remainder of $\displaystyle 30 \cdot 22 + 30$ when divided by 7.
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Problem 2
Find the least nonnegative remainder of $\displaystyle 35 \cdot 26 + 35$ when divided by 8.
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Problem 3
Find the least nonnegative remainder of $\displaystyle 37 \cdot 30 + 37$ when divided by 9.
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Problem 4
Find the least nonnegative remainder of $\displaystyle 42 \cdot 34 + 42$ when divided by 10.
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Problem 5
Find the least nonnegative remainder of $\displaystyle 47 \cdot 34 + 47$ when divided by 11.
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Problem 6
Find the least nonnegative remainder of $\displaystyle 49 \cdot 38 + 49$ when divided by 12.
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Problem 7New context
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?
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Problem 8New context
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?
Practise this exact problem →

Open stage NT 2.3 in the student workspace →

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Factorials and Representations

Prime Powers in Factorials

Understand the idea

Choose and carry out a method

Check the reasoning

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

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

Common pitfalls

Explain it to yourself

Place Value in Other Bases

Understand the idea

Choose and carry out a method

Check the reasoning

Convert the base-6 numeral 123 to base ten.

Convert the base-7 numeral 123 to base ten.

Common pitfalls

Explain it to yourself

Computing with Remainders

Understand the idea

Choose and carry out a method

Check the reasoning

Find the least nonnegative remainder of 18⋅14+18\displaystyle 18 \cdot 14 + 1818⋅14+18 when divided by 4.

Find the least nonnegative remainder of 23⋅18+23\displaystyle 23 \cdot 18 + 2323⋅18+23 when divided by 5.

Common pitfalls

Explain it to yourself

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

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

Find the least nonnegative remainder of $\displaystyle 18 \cdot 14 + 18$ when divided by 4.

Find the least nonnegative remainder of $\displaystyle 23 \cdot 18 + 23$ when divided by 5.