Modular Arithmetic PDF

SLIDESMANIA.COM Modular Arithmetic And its Applications  MODULAR ARITHMETIC Related to that of the remainder in division. The operation of finding the remainder is sometimes referred...

SLIDESMANIA.COM Modular Arithmetic And its Applications  MODULAR ARITHMETIC Related to that of the remainder in division. The operation of finding the remainder is sometimes referred to as the modulo operation. SLIDESMANIA.COM  Time Supposed it’s 10:00 PM and you set your alarm for tomorrow morning to attend your class. As you set the time the notification shows that “ the alarm will remind you in 7 hours” What is the time that you have set? SLIDESMANIA.COM  Time What is 14 hours after 4:00 AM? To determine what will be the time 14 hours after 4:00 AM, we must divide 14 by 12, because for every 12 hours, the short hand of the clock would be on the same number on the clock again. 14 ÷ 12 = 1 𝑟. 2 The hand of the clock passed through 12 only once, therefore as the hand of the SLIDESMANIA.COM clock returns to 4 after 12 hours, from AM, it would become PM. 12 hours from 4AM is 4PM.  Time What is 14 hours after 4:00 AM? Now we have the remainder of 2. We already know that 12 hours after 4AM is 4PM. Just add 2 more hours 4𝑃𝑀 + 2 ℎ𝑜𝑢𝑟𝑠 = 6𝑃𝑀 SLIDESMANIA.COM Therefore, 14 hours after 4AM is 6:00PM.  Time What is 100 hours before 1:00 PM? Note that it is aking for the “before hour” Simply divide 100 by 12. 100 ÷ 12 = 8 𝑟. 4 SLIDESMANIA.COM  Time What is 100 hours before 1:00 PM? Number of 12 hours Shift of AM and From the given 4AM, the shift of (Quotient) PM AM and PM depends upon the 1 PM number of 12 hours or the quotient. If the quotient is ODD, 2 AM the shift of the given will 3 PM CHANGE (from AM it becomes 4 AM PM) and if the quotient is EVEN, the shift of the given will be the 5 PM SAME. SLIDESMANIA.COM 6 AM  Time What is 100 hours before 1:00 PM? 100 ÷ 12 = 8 𝑟. 4 Since the quotient is 8 which is an EVEN number, therefore the reference point for the answer would be the SAME of the shift of the given which is PM. Moving 4 hours backward from 1 PM, 1PM - 12PM - 11AM - 10AM - 9AM - 8AM SLIDESMANIA.COM  Time What is 247 hours after 3:00 PM? Simply divide 247 by 12. 247 ÷ 12 = 20 𝑟. 7 Since the quotient is 20, which is even, the Quotient Shift (Number of 12 (Reference time for reference time will be the SAME. hours) the answer) Then, add the remainder, 7 hours, to 3PM: Even Same 3 PM + 7 hours = 10 PM SLIDESMANIA.COM Odd Change  Time The quotient after dividing by 12 the given number of hours will tell us whether the reference time for the answer (shift) will be the same or it will be changed. The remainder will be the number of hours to be added (after) and to be subtracted (before) from the reference time. SLIDESMANIA.COM  Day If today is Monday, what day is 16 days from today? In determining a particular day in a week, our divisor will be 7 because we have 7 days in a week. Simply divide 16 by 7, 𝟏𝟔 ÷ 𝟕 = 𝟐 𝒓. 𝟐 We add the remainder to the reference day, Monday Tuesday Wednesday SLIDESMANIA.COM + 2 days  Day If today is Saturday, what was the day 211 days ago? Simply divide 211 by 7, 𝟐𝟏𝟏 ÷ 𝟕 = 𝟑𝟎 𝒓. 𝟏 Take note of the keyword used. It will tell us whether we’re going to add or subtract the remainder from the reference day. The keyword is “ago”. We will move 1 day backward from Saturday, SLIDESMANIA.COM Friday  Day What day will it be after 147 days if today is Sunday? Simply divide 147 by 7, 𝟏𝟒𝟕 ÷ 𝟕 = 𝟐𝟏 𝒓. 𝟎 Sunday SLIDESMANIA.COM  Date In a year, a day will automatically move one day on the following year. For example, If your birthday this year falls on a Monday, next year your birthday will fall on a Tuesday. Nov. 13, 2021 Nov. 13, 2022 MONDAY TUESDAY But if the following year is a LEAP year there may be a need for an extra move SLIDESMANIA.COM of day/s.  Date If May 19, 2013 was Thursday, on what day will it fall in the year 2022? Determine how many years are there in the interval from 2013 and 2022. 2022-2013 = 9 Automatic Move Determine how many leap years are in the interval 2016 and 2020 = 2 leap years Extra Move SLIDESMANIA.COM  Date If May 19, 2013 was Thursday, on what day will it fall in the year 2022? 9 𝑎𝑢𝑡𝑜𝑚𝑎𝑡𝑖𝑐 𝑚𝑜𝑣𝑒𝑠 + 2 𝑒𝑥𝑡𝑟𝑎 𝑚𝑜𝑣𝑒𝑠 = 11 Do the same process in finding specific day, 11 ÷ 7 = 1 𝑟. 4 SLIDESMANIA.COM Adding 4 days from Thursday, so the answer is Monday.  Day If February 14, 2021 will fall on a Sunday, on what day will it fall in the year 2036? Determine how many years are there in the interval from 2013 and 2022. 2036-2021 = 15 Automatic Move Determine how many leap years are in the interval 2024, 2028, 2032, and 2036 = 4 leap years Dates When they will be having their extra move? 2025, 2029, and 2033. SLIDESMANIA.COM January 1 – February 28 The year after the leap year 3 March 11 – December 31 Leap Year Extra Move  Day If February 14, 2021 will fall on a Sunday, on what day will it fall in the year 2036? 15 𝑎𝑢𝑡𝑜𝑚𝑎𝑡𝑖𝑐 𝑚𝑜𝑣𝑒𝑠 + 3 𝑒𝑥𝑡𝑟𝑎 𝑚𝑜𝑣𝑒𝑠 = 18 Do the same process in finding specific day, 18 ÷ 7 = 2 𝑟. 4 SLIDESMANIA.COM Adding 4 days from Sunday, so the answer is Thursday.  Day In 2021, we celebrated Independence day on the second Friday of June. On what day will Independence day fall on 2041? Determine how many years are there in the interval from 2041 to 2021. 2041-2021 = 20 Automatic Move Determine how many leap years are in the interval 2024, 2028, 2032, 2036, and 2040 = 5 leap years Extra Move SLIDESMANIA.COM  Day This year, we celebrated Independence day on the second Friday of June. On what day will Independence day fall on 2041? 20 𝑎𝑢𝑡𝑜𝑚𝑎𝑡𝑖𝑐 𝑚𝑜𝑣𝑒𝑠 + 5 𝑒𝑥𝑡𝑟𝑎 𝑚𝑜𝑣𝑒𝑠 = 25 Do the same process in finding specific day, 25 ÷ 7 = 3 𝑟. 4 SLIDESMANIA.COM Adding 4 days from Friday, so the answer is Tuesday.  The Modulo Congruence Two integers a and b are said to be congruent modulo n, 𝑎−𝑏 where n is a natural number, if 𝑛 is an integer. 𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏 Modulus Congruence SLIDESMANIA.COM  TRUE OR FALSE 𝟏𝟏 ≡ 𝟏 𝒎𝒐𝒅 𝟐 TRUE 𝟏𝟏−𝟏 because 𝟐 =𝟓 and 5 is an integer Two integers a and b are said to be congruent 𝒂−𝒃 modulo n, where n is a natural number, if 𝒏 is an integer. SLIDESMANIA.COM 𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏  TRUE OR FALSE 𝟐𝟓 ≡ 𝟑 𝒎𝒐𝒅 𝟕 FALSE 𝟐𝟓−𝟑 𝟏 because 𝟕 = 𝟑 𝟕 𝟏 Two integers a and b are said to be congruent and 𝟑 𝟕 is NOT an integer 𝒂−𝒃 modulo n, where n is a natural number, if 𝒏 is an integer. SLIDESMANIA.COM 𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏  TRUE OR FALSE 𝟒𝟖 ≡ 𝟑 𝒎𝒐𝒅 𝟓 Two integers a and b are said to be congruent 𝒂−𝒃 TRUE modulo n, where n is a natural number, if 𝒏 is an integer. SLIDESMANIA.COM 𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏  TRUE OR FALSE 𝟕𝟔 ≡ 𝟔 𝒎𝒐𝒅 𝟕 Two integers a and b are said to be congruent 𝒂−𝒃 TRUE modulo n, where n is a natural number, if 𝒏 is an integer. SLIDESMANIA.COM 𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏  TRUE OR FALSE 𝟓𝟗 ≡ 𝟖 𝒎𝒐𝒅 𝟗 Two integers a and b are said to be congruent 𝒂−𝒃 FALSE modulo n, where n is a natural number, if 𝒏 is an integer. SLIDESMANIA.COM 𝒂 ≡ 𝒃 𝒎𝒐𝒅 𝒏  Barcodes Here’s a barcode of a 1.5L of Sprite. Solve for the missing digit. 4 801981 11860X 4 801981 11860X 𝑑13 = 10 − 𝑑1 + 3𝑑2 + 𝑑3 + 3𝑑4 + 𝑑5 + 3𝑑6 + 𝑑7 + 3𝑑8 + 𝑑9 + 3𝑑10 + 𝑑11 + 3𝑑12 𝑚𝑜𝑑 10 𝑑13 = 10 − 4 + 24 + 0 + 3 + 9 + 24 + 1 + 3 + 1 + 24 + 6 + 0 𝑚𝑜𝑑 10 𝑑13 = 10 − (99) 𝑚𝑜𝑑 10 𝑑13 = 10 − 9 𝒅𝟏𝟑 = 𝟏 SLIDESMANIA.COM 𝑑13 = 10 − 𝑑1 + 3𝑑2 + 𝑑3 + 3𝑑4 + 𝑑5 + 3𝑑6 + 𝑑7 + 3𝑑8 + 𝑑9 + 3𝑑10 + 𝑑11 + 3𝑑12 𝑚𝑜𝑑 10  Barcodes Here’s a barcode of Oishi Prawn Crackers. Solve for the missing digit. 4 80019X 177962 4 80019X 177962 2 = 10 − 4 + 24 + 0 + 0 + 1 + 27 + 𝑋 + 3 + 7 + 21 + 9 + 18 𝑚𝑜𝑑 10 2 = 10 − (𝑋 + 114) 𝑚𝑜𝑑 10 𝑋 + 114 𝑚𝑜𝑑 10 = 10 − 2 𝑋 + 114 𝑚𝑜𝑑 10 = 8 4 + 114 𝑚𝑜𝑑 10 = 8 𝑻𝒉𝒆 𝟕𝒕𝒉 𝒅𝒊𝒈𝒊𝒕 𝒊𝒔 𝟒. SLIDESMANIA.COM 𝑑13 = 10 − 𝑑1 + 3𝑑2 + 𝑑3 + 3𝑑4 + 𝑑5 + 3𝑑6 + 𝑑7 + 3𝑑8 + 𝑑9 + 3𝑑10 + 𝑑11 + 3𝑑12 𝑚𝑜𝑑 10  Thanks! If you have any questions, feel free to contact me on the contact details I have provided you. SLIDE CREDITS: Ms. FATIMA PEREZ SLIDESMANIA.COM  PROBLEM SET # 3 1. What is 1423 hours before and after 11:00 AM? 2. What is 10234 hours before and after 2:00 PM? 3. If today is Tuesday, what day is 161 days from today? 4. If today is Friday, what was the day 226 days ago? 5. If March 21, 2021, will fall on a Sunday, on what day will it fall in the year 2051? 6. If January 17, 2011, will fall on a Friday, on what day will it fall in the year 2067? SLIDESMANIA.COM