Last Two Digit of a Number raised to Power
Last Two Digit of the Number is basically a Tens Digit and Unit Digit of the number.To understand the things in detail, let us take an example .
Example : The last two digits of the number, 1453289 is 8 and 9 ; which is quite obvious.
To calculate the last two digit of a Number raised to power
Case 1: Number ending with 1.
Method : For the Numbers ending with 1,
abc1^(xyz)
Unit Digit = 1
Tens Digit = c * z
(c * z) 1
The last two digits of the Number; abc1^(xyz) = ━━━ ━━━
↓ ↓
Tens Digit Unit Digit
Example : Calculate the Last two Digits of 654921⁶⁷⁵⁴⁸ .
Sol : Since the digit last with 21 .
Unit Digit = 1
For Calculating the Tens Digit = 2 x 8 = 16
Hence, the Last two Digit = 6 1
━ ━
Example : Calculate the Last Two Digits of 61341⁸¹⁵
Sol : Since the digit last with 41 .
Unit Digit = 1
For Calculating the Tens Digit = 4 x 5 = 20
Hence, the Last two Digit = 0 1
━ ━
Case 2: Number ending with 3, 7 or 9.
Method : If the number ends with 3 → 3⁴ → Last Digit = 1
If the number ends with 7 → 7⁴ → Last Digit = 1
If the number ends with 9 → 9² → Last Digit = 1
Example : Calculate the last Digit of 69³².
Sol : If the number ends with 9 → 9² → Last Digit = 1
Square the base and Divide the exponent by 2.
(69²)¹⁶ = Last to digits of 69² = 61 = (61)¹⁶
Since the digit last with 61 .
Unit Digit = 1
For Calculating the Tens Digit = 6 x 6 = 36
Hence, the Last two Digit = 6 1
━ ━
Example : Calculate the last two digits of 37⁶⁴.
Sol : If the number ends with 7 → 7⁴ → Last Digit = 1
Raise the base by 4 and Divide the exponent by 4.
(37⁴)¹⁶ = Last two digits of 37⁴ = 61 = (61)¹⁶
Since the digit last with 61 .
Unit Digit = 1
For Calculating the Tens Digit = 6 x 6 = 36
Hence, the Last two Digit = 6 1
━ ━
Case 3: Number ending with 2, 4 , 6 or 8
Method : The last two digits for the number ending with 2, 4, 6 or 8 is :
(2)¹⁰ → 1024
Last two digits are 24.
(24)^odd number → Last two digit = 24
(24)^even number → Last two digit = 76
76 x 2^any power → Last two digit = 2^any power
Example : Calculate the last two digits of 2¹²⁸.
Sol : 2¹²⁸ = 2¹²⁰ x 2⁸ = (2¹⁰)¹² x 2⁸
= (1024)¹² x 2⁸ [ (24)^even number → Last two digit = 76 ]
= 76 x 56
= 56
Hence , the last two digits of 2¹²⁸ = 56
Example : Calculate the last two digits of 4⁴⁰ .
Sol : 4⁴⁰ = (2²)⁴⁰ = 2⁸⁰
= (2¹⁰)⁸
= (1024)⁸ [ (24)^even number → Last two digit = 76 ]
= 76
Hence, the last two digits of 4⁴⁰ = 76
Case 4: Number ending with 5
Method :
[ (1/3/5/7/9) 5 ]^ ODD → Last two digit = 75
━━━━━ ━
↓ ↓
Tens Digit Unit Digit
[ (2/4/6/8/0) 5 ]^ ODD → Last two digit = 25
━━━━━ ━
↓ ↓
Tens Digit Unit Digit
[Any Value 5 ]^ EVEN → Last two digit = 25
━━━━━ ━
↓ ↓
Tens Digit Unit Digit
Example : Find the last two digits of 185⁸⁸.
Sol : [Any Value 5 ]^ EVEN → Last two digit = 25
━━━━━ ━
↓ ↓
Tens Digit Unit Digit
85⁸⁸ = Last two Digit = 25.
Hence, the last two digits of 185⁸⁸ = 25
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