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Chapter 4: Problem 2

Prove the following statements: (a) For any integer \(a\), the units digit of \(a^{2}\) is \(0,1,4,5,6\), or 9 . (b) Any one of the integers \(0,1,2,3,4,5,6,7,8,9\) can occur as the units digit of \(a^{3}\). (c) For any integer \(a\), the units digit of \(a^{4}\) is \(0,1,5\), or 6 . (d) The units digit of a triangular number is \(0,1,3,5,6\), or 8 .

Short Answer

Expert verified
(a) 0,1,4,5,6,9; (b) 0-9; (c) 0,1,5,6; (d) 0,1,3,5,6,8.

Step by step solution

01

Understanding Units Digits in Powers

When calculating the units digit of a power, we focus on the units digit of the base number due to the cyclic nature of arithmetic modulo 10. The units digit of the result depends only on the units digit of the base.
02

Square Numbers Modulo 10

For any integer, consider the units digit of \(a\) as "0, 1, 2, 3, 4, 5, 6, 7, 8, 9". Calculate \(a^2\) for each digit.- \(0^2 = 0\)- \(1^2 = 1\)- \(2^2 = 4\)- \(3^2 = 9\)- \(4^2 = 16\) (whose units digit is 6)- \(5^2 = 25\) (whose units digit is 5)- \(6^2 = 36\) (whose units digit is 6)- \(7^2 = 49\) (whose units digit is 9)- \(8^2 = 64\) (whose units digit is 4)- \(9^2 = 81\) (whose units digit is 1)The possible units digits for \(a^2\) are 0, 1, 4, 5, 6, 9.
03

Cube Numbers Modulo 10

Calculate \(a^3\) for units digits "0, 1, 2, 3, 4, 5, 6, 7, 8, 9":- \(0^3 = 0\)- \(1^3 = 1\)- \(2^3 = 8\)- \(3^3 = 27\) (whose units digit is 7)- \(4^3 = 64\) (whose units digit is 4)- \(5^3 = 125\) (whose units digit is 5)- \(6^3 = 216\) (whose units digit is 6)- \(7^3 = 343\) (whose units digit is 3)- \(8^3 = 512\) (whose units digit is 2)- \(9^3 = 729\) (whose units digit is 9)The possible units digits for \(a^3\) are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
04

Fourth Powers Modulo 10

Calculate \(a^4\) for units digits "0, 1, 2, 3, 4, 5, 6, 7, 8, 9":- \(0^4 = 0\)- \(1^4 = 1\)- \(2^4 = 16\) (whose units digit is 6)- \(3^4 = 81\) (whose units digit is 1)- \(4^4 = 256\) (whose units digit is 6)- \(5^4 = 625\) (whose units digit is 5)- \(6^4 = 1296\) (whose units digit is 6)- \(7^4 = 2401\) (whose units digit is 1)- \(8^4 = 4096\) (whose units digit is 6)- \(9^4 = 6561\) (whose units digit is 1)The possible units digits for \(a^4\) are 0, 1, 5, 6.
05

Units Digit of Triangular Numbers

Triangular numbers can be expressed as \( \frac{n(n+1)}{2} \). Each triangular number is either \(n\) or \(n+1\) being even while the other is odd. Analyse the series of units digits for triangular numbers as \(n\) goes from 1 to 10:- \(T_1 = 1\)- \(T_2 = 3\)- \(T_3 = 6\)- \(T_4 = 10\) (whose units digit is 0)- \(T_5 = 15\) (whose units digit is 5)- \(T_6 = 21\) (whose units digit is 1)- \(T_7 = 28\) (whose units digit is 8)- \(T_8 = 36\) (whose units digit is 6)- \(T_9 = 45\) (whose units digit is 5)- \(T_{10} = 55\) (whose units digit is 5)These cover all possibilities of 0, 1, 3, 5, 6, 8.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Numbers Modulo 10
The units digit of a square number is determined by taking the units digit of the number being squared and raising it to the power of 2. When analyzing square numbers modulo 10, you observe a pattern because the result is influenced by the units digit of the base number. Here are the important steps to understand this concept:

  • The units digit of 0 squared (\(0^2\)) is 0.
  • The units digit of 1 squared (\(1^2\)) is 1.
  • The units digit of 2 squared (\(2^2\)) is 4.
  • The units digit of 3 squared (\(3^2\)) is 9.
  • The units digit of 4 squared (\(4^2\)) is 6.
  • The units digit of 5 squared (\(5^2\)) is 5.
  • The units digit of 6 squared (\(6^2\)) is 6.
  • The units digit of 7 squared (\(7^2\)) is 9.
  • The units digit of 8 squared (\(8^2\)) is 4.
  • The units digit of 9 squared (\(9^2\)) is 1.
So, when calculating the square of any integer, the possible units digits are 0, 1, 4, 5, 6, or 9.
Cube Numbers Modulo 10
To find the units digit of a cube number, you focus again on the units digit of the original number and determine the result by cubing it. Each units digit of a number from 0 to 9 gives a distinct result when cubed:

  • 0 cubed (\(0^3\)) gives a units digit of 0.
  • 1 cubed (\(1^3\)) remains 1.
  • 2 cubed (\(2^3\)) becomes 8.
  • 3 cubed (\(3^3\)) results in 7.
  • 4 cubed (\(4^3\)) ends in 4.
  • 5 cubed (\(5^3\)) yields 5.
  • 6 cubed (\(6^3\)) leads to 6.
  • 7 cubed (\(7^3\)) ends with 3.
  • 8 cubed (\(8^3\)) gives 2.
  • 9 cubed (\(9^3\)) is 9.
As noted, when dealing with cube numbers, the units digit can be any digit between 0 to 9, as there is no restriction on the possible outcomes.
Fourth Powers Modulo 10
When exploring fourth powers modulo 10, you essentially look at what happens when the units digits of numbers from 0 to 9 are raised to the power of 4. Fourth powers reveal a restricted set of possible units digits:

  • 0 to the power of 4 (\(0^4\)) yields a units digit of 0.
  • 1 to the power of 4 (\(1^4\)) is 1.
  • 2 to the power of 4 (\(2^4\)) ends with 6.
  • 3 to the power of 4 (\(3^4\)) results in 1.
  • 4 to the power of 4 (\(4^4\)) gives 6.
  • 5 to the power of 4 (\(5^4\)) is 5.
  • 6 to the power of 4 (\(6^4\)) results in 6.
  • 7 to the power of 4 (\(7^4\)) ends with 1.
  • 8 to the power of 4 (\(8^4\)) concludes with 6.
  • 9 to the power of 4 (\(9^4\)) finishes with 1.
Thus, the units digit of any fourth power is always 0, 1, 5, or 6.
Triangular Numbers
Triangular numbers are a fascinating sequence of numbers, calculated by the formula: \( T_n = \frac{n(n+1)}{2} \). This sequence involves summing consecutive integers starting from 1, forming numbers which, if you were to use dots to illustrate them, would resemble triangles.

The units digits of triangular numbers repeat a certain pattern. Let's look at the first few examples:

  • The 1st triangular number is 1, with a units digit of 1.
  • The 2nd triangular number is 3, with a units digit of 3.
  • The 3rd triangular number is 6, with a units digit of 6.
  • The 4th triangular number is 10, ending with 0.
  • The 5th triangular number is 15, ending with 5.
  • The 6th triangular number is 21, with a units digit of 1.
  • The 7th triangular number is 28, finishing with 8.
  • The 8th triangular number is 36, ending with 6.
  • The 9th triangular number is 45, concluding with 5.
  • The 10th triangular number is 55, also finishing with 5.
From this, you can derive the possible units digits for triangular numbers as 0, 1, 3, 5, 6, or 8, giving these numbers a unique identity among numeric sequences.

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Most popular questions from this chapter

Use the binary exponentiation algorithm to compute both \(19^{53}(\bmod 503)\) and \(141^{47}\) \((\bmod 1537)\)

Establish that if \(a\) is an odd integer, then for any \(n \geq 1\) $$ a^{x}=1\left(\bmod 2^{n+2}\right) $$ [Hint: Proceed by induction on \(n\).

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