Question 1 - Figure it out - Page 44, 45 - Chapter 2 Class 8 - Power Play (Ganita Prakash)

Transcript

Question 1 Find out the units digit in the value of 2224 ÷ 432? [Hint: 4 = 22]Let’s simplify it 2224 ÷ 432 = 2^224/4^32 Writing 4 = 22 = 2^224/〖(𝟐^𝟐)〗^𝟑𝟐 Using (𝒂^𝒑 )^𝒒=𝒂^(𝒑 × 𝒒) = 2^224/𝟐^(𝟐 × 𝟑𝟐) = 2^224/2^64 Using 𝒂^𝒑/𝒂^𝒒 =𝒂^(𝒑 − 𝒒) = 2^(224−64) = 𝟐^𝟏𝟔𝟎 Now, we need to find Unit digit of 𝟐^𝟏𝟔𝟎. For that first lets find pattern of Powers of 2 Unit digit Pattern for Powers of 2 Powers of 2 are 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 Thus, the pattern of the units digits is 2, 4, 8, 6, which is a cycle of 4. Pattern is If power divided by 4 has remainder 1, then unit digit is 2 If power divided by 4 has remainder 2, then unit digit is 4 If power divided by 4 has remainder 3, then unit digit is 8 If power divided by 4 has remainder 0, then unit digit is 6 Finding Unit digit of 𝟐^𝟏𝟔𝟎 Diving Power 160 by 4, We get remainder 0 Thus, Unit digit of 2^160 is 6 So, required unit digit is 6