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(a) Start by squaring both sides of the congruence 34 = 81 (mod 100) to prove that 38 = 61 (mod 100) and then prove that 3¹6 = 21 (mod 100). What does this tell you about the last two digits in the decimal repre- sentation of 316? (b) Use the two congruences in Part (24a) and laws of exponents to deter- mine r where 320 = r (mod 100) and r € Z with 0 < r < 100. What does this tell you about the last two digits in the decimal representation of 320? 158 @080 BY NC SA Chapter 3. Constructing and Writing Proofs in Mathematics (c) Determine the last two digits in the decimal representation of 3400 (d) Determine the last two digits in the decimal representation of 4804. Hint: One way is to determine the "mod 100 values” for 4², 44, 48, 416, 432, 464, and so on. Then use these values and laws of exponents to determine r, where 4804 = r (mod 100) and r € Z with 0 < r < 100. (e) Determine the last two digits in the decimal representation of 33356 (f) Determine the last two digits in the decimal representation of 7403

(a) Start by squaring both sides of the congruence 34 = 81 (mod 100) to prove that 38 = 61 (mod 100) and then prove that 3¹6 = 21 (mod 100). What does this tell you about the last two digits in the decimal repre- sentation of 316? (b) Use the two congruences in Part (24a) and laws of exponents to deter- mine r where 320 = r (mod 100) and r € Z with 0 < r < 100. What does this tell you about the last two digits in the decimal representation of 320? 158 @080 BY NC SA Chapter 3. Constructing and Writing Proofs in Mathematics (c) Determine the last two digits in the decimal representation of 3400 (d) Determine the last two digits in the decimal representation of 4804. Hint: One way is to determine the "mod 100 values” for 4², 44, 48, 416, 432, 464, and so on. Then use these values and laws of exponents to determine r, where 4804 = r (mod 100) and r € Z with 0 < r < 100. (e) Determine the last two digits in the decimal representation of 33356 (f) Determine the last two digits in the decimal representation of 7403

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**The Last Two Digits of a Large Integer** Notice that \(7, 381, 272 \equiv 72 \, (\text{mod} \, 100)\) since \(7, 381, 272 - 72 = 7, 381, 200\), which is divisible by 100. In general, if we start with an integer whose decimal representation has more than two digits and subtract the integer formed by the last two digits, the result will be an integer whose last two digits are 00. This result will be divisible by 100. Hence, any integer with more than 2 digits is congruent modulo 100 to the integer formed by its last two digits. (a) Start by squaring both sides of the congruence \(3^{4} \equiv 81 \, (\text{mod} \, 100)\) to prove that \(3^{8} \equiv 61 \, (\text{mod} \, 100)\) and then prove that \(3^{16} \equiv 21 \, (\text{mod} \, 100)\). What does this tell you about the last two digits in the decimal representation of \(3^{16}\)? (b) Use the two congruences in Part (24a) and laws of exponents to determine \(r\) where \(3^{20} \equiv r \, (\text{mod} \, 100)\) and \(r \in \mathbb{Z}\) with \(0 \le r < 100\). What does this tell you about the last two digits in the decimal representation of \(3^{20}\)? (c) Determine the last two digits in the decimal representation of \(3^{400}\). (d) Determine the last two digits in the decimal representation of \(4^{804}\). *Hint: One way is to determine the “mod 100 values” for \(4^{2}, 4^{4}, 4^{8}, 4^{16}, 4^{32}, 4^{64}\), and so on. Then use these values and laws of exponents to determine \(r\), where \(4^{804} \equiv r \, (\text{mod} \, 100)\) and \(r \in \mathbb{Z}\) with \(0 \le r < 100\).* (e)
Transcribed Image Text:**The Last Two Digits of a Large Integer** Notice that \(7, 381, 272 \equiv 72 \, (\text{mod} \, 100)\) since \(7, 381, 272 - 72 = 7, 381, 200\), which is divisible by 100. In general, if we start with an integer whose decimal representation has more than two digits and subtract the integer formed by the last two digits, the result will be an integer whose last two digits are 00. This result will be divisible by 100. Hence, any integer with more than 2 digits is congruent modulo 100 to the integer formed by its last two digits. (a) Start by squaring both sides of the congruence \(3^{4} \equiv 81 \, (\text{mod} \, 100)\) to prove that \(3^{8} \equiv 61 \, (\text{mod} \, 100)\) and then prove that \(3^{16} \equiv 21 \, (\text{mod} \, 100)\). What does this tell you about the last two digits in the decimal representation of \(3^{16}\)? (b) Use the two congruences in Part (24a) and laws of exponents to determine \(r\) where \(3^{20} \equiv r \, (\text{mod} \, 100)\) and \(r \in \mathbb{Z}\) with \(0 \le r < 100\). What does this tell you about the last two digits in the decimal representation of \(3^{20}\)? (c) Determine the last two digits in the decimal representation of \(3^{400}\). (d) Determine the last two digits in the decimal representation of \(4^{804}\). *Hint: One way is to determine the “mod 100 values” for \(4^{2}, 4^{4}, 4^{8}, 4^{16}, 4^{32}, 4^{64}\), and so on. Then use these values and laws of exponents to determine \(r\), where \(4^{804} \equiv r \, (\text{mod} \, 100)\) and \(r \in \mathbb{Z}\) with \(0 \le r < 100\).* (e)
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