Question 3 of 28advanced🔍 AnalyzeNumerical3 marks

Find out the units digit in the value of 2224÷4322^{224} \div 4^{32}? [Hint: 4=224 = 2^2]

Correct Answer

Given: 2224÷4322^{224} \div 4^{32}

Hint: 4=224 = 2^2

Step 1: Rewrite 4324^{32} using the hint

432=(22)32=2644^{32} = (2^2)^{32} = 2^{64}

Step 2: Apply division rule for exponents

2224÷264=2224−64=21602^{224} \div 2^{64} = 2^{224-64} = 2^{160}

Step 3: Find the pattern of units digits for powers of 2 21=22^1 = 2, 22=42^2 = 4, 23=82^3 = 8, 24=162^4 = 16 (units digit 6), 25=322^5 = 32 (units digit 2) The cycle repeats every 4 powers.

Step 4: Determine the position in the cycle 160÷4=40160 \div 4 = 40 with remainder 0 ∴ 21602^{160} has the same units digit as 242^4

Step 5: The units digit of 24=162^4 = 16 is 6

Answer: The units digit is 6.

Exercise: Figure it Out (Section 2) | Q: 1 | (Page Page 26)
For More Understanding

Explanation

Using the hint that 4=224 = 2^2, we can rewrite 432=(22)32=2644^{32} = (2^2)^{32} = 2^{64}. Then 2224÷432=2224÷264=2224−64=21602^{224} \div 4^{32} = 2^{224} \div 2^{64} = 2^{224-64} = 2^{160}. The units digit of powers of 2 follows a pattern: 21=22^1=2, 22=42^2=4, 23=82^3=8, 24=162^4=16 (units digit 6), 25=322^5=32 (units digit 2), and the cycle repeats every 4 powers. Since 160÷4=40160 \div 4 = 40 with remainder 0, 21602^{160} has the same units digit as 242^4, which is 6.

Solution Steps

  1. Step 1: Rewrite 4324^{32} using 4=224 = 2^2: 432=(22)32=2644^{32} = (2^2)^{32} = 2^{64}

  2. Step 2: Apply division rule for exponents: 2224÷264=2224−64=21602^{224} \div 2^{64} = 2^{224-64} = 2^{160}

  3. Step 3: Find the pattern of units digits for powers of 2: 21=22^1=2, 22=42^2=4, 23=82^3=8, 24=162^4=16 (units digit 6)

  4. Step 4: The pattern repeats every 4 powers. Since 160÷4=40160 \div 4 = 40 with remainder 0, 21602^{160} has the same units digit as 242^4

  5. Step 5: Therefore, the units digit is 6

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