Question

The units digit of the expression 23^22*565^46*91^43

Answer 1

Answer:

unit digit of some nubers is always same like 5 and6 and 1

Answer 2

• The units digit of the expression (23²² × 565⁴⁶ × 91⁴³) is equal to 5 .

To Find :- The units digit of the expression :-

(23^22) × (565^46) × (91^43) .

Concept used :-

• Unit digit of a number depends on the last digit digit of the number .

Solution :-

we know that,

→ 3¹ = 3 = Unit digit 3

→ 3² = 9 = Unit digit 9

→ 3³ = 27 = Unit digit 7

→ 3⁴ = 81 = Unit digit 1

So, unit digit of 3 to the power repeats after that .

Conclusion :-

• 3^(4n + 1) = Unit digit 3
• 3^(4n + 2) = Unit digit 9
• 3^(4n + 3) = Unit digit 7
• 3^(4n + 4) = unit digit 1
• where n = whole numbers .

then,

→ (23)²²

since unit digit will depends on the unit digit of 23 that is 3 .

→ (3)²²

→ (3)^(4×5 + 2)

comparing with 3^(4n + 2) we get,

→ unit digit 9 .

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Now, we know that,

→ 5¹ = 5 = Unit digit 5

→ 5² = 25 = Unit digit 5

→ 5³ = 125 = Unit digit 5

→ 5⁴ = 625 = Unit digit 5

Conclusion :-

• 5^n = unit digit 5 .
• Where n = Natural numbers .

So,

→ (565)⁴⁶

since unit digit will depends on the unit digit of 565 that is 5.

then,

→ (5)⁴⁶

→ Unit digit 5 .

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Now, we know that,

→ 1¹ = 1 = Unit digit 1

→ 1² = 1 = Unit digit 1

→ 1³ = 1 = Unit digit 1

→ 1⁴ = 1 = Unit digit 1

Conclusion :-

• 1^n = unit digit 1 .
• Where n = Whole numbers .

So,

→ (91)⁴³

since unit digit will depends on the unit digit of 91 that is 1.

then,

→ (1)⁴³

→ Unit digit 1 .

therefore,

→ Unit digit [23²² × 565⁴⁶ × 91⁴³]

→ (Unit digit 23²²) × (Unit digit 565⁴⁶) × (Unit digit 91⁴³)

→ 9 × 5 × 1

→ 45

→ Unit digit 5 (Ans.)

Hence, Required units digit of the given expression is equal to 5 .

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