The Last Two Digits of a Large Integer.Notice that 7, 381, 272 = 72 (mod 100) since 7, 381, 272-72= 7, 381, 200,which is divisible by 100. In general, if we start with an integer whose deci-mal representation has more than two digits and subtract the integer formedby the last two digits, the result will be an integer whose last two digits are00. This result will be divisible by 100. Hence, any integer with more than2 digits is congruent modulo 100 to the integer formed by its last two digits.(a) Start by squaring both sides of the congruence 34 = 81 (mod 100) toprove 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

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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