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Zorluk: ZorUnit Digit and Cyclicity

What is the unit digit of the composite exponential expression E=(432316×657235)+875432−959411E = (432^{316} \times 657^{235}) + 875^{432} - 959^{411}?

  1. 4Cevap
  2. B
    9
  3. C
    6
  4. D
    0

Cevap

The unit digit of the expression is 4.
Evaluating each term using cyclicity: 432316432^{316} ends in 6 (242^4), 657235657^{235} ends in 3 (737^3), 875432875^{432} ends in 5 (5n5^n), and 959411959^{411} ends in 9 (9odd9^{\text{odd}}). Combining these gives (6×3)+5−9=18+5−9=14(6 \times 3) + 5 - 9 = 18 + 5 - 9 = 14, yielding a unit digit of 4.

Adım Adım Çözüm

1
Determine the unit digit of 432316432^{316}
Unit digit is 6
The unit digit of the base is 2, which has a cyclicity pattern of 4 (2, 4, 8, 6). The exponent 316 is divisible by 4 with remainder 0. For a remainder of 0, we take the 4th power in the cycle: 24=162^4 = 16, giving a unit digit of 6.
2
Determine the unit digit of 657235657^{235}
Unit digit is 3
The unit digit of the base is 7, which has a cyclicity pattern of 4 (7, 9, 3, 1). Dividing the exponent 235 by 4 gives a remainder of 3 (235=4×58+3235 = 4 \times 58 + 3). We take the 3rd power in the cycle: 73=3437^3 = 343, giving a unit digit of 3.
3
Calculate the unit digit of the product (432316×657235)(432^{316} \times 657^{235})
Unit digit is 8
Multiplying the unit digits of the two terms yields 6×3=186 \times 3 = 18, which has a unit digit of 8.
4
Determine the unit digit of 875432875^{432} and add it to the product
Unit digit of sum is 3
Any positive integer power of a base ending in 5 always ends in 5. Adding this to the product's unit digit gives 8+5=138 + 5 = 13, so the sum ends in 3.
5
Determine the unit digit of 959411959^{411} and subtract it to find the final unit digit
Final unit digit is 4
The unit digit of the base is 9, which has a cyclicity of 2 (9 for odd powers, 1 for even powers). Since 411 is odd, 959411959^{411} ends in 9. Subtracting this from the sum's unit digit gives 3−9≡13−9=43 - 9 \equiv 13 - 9 = 4.

Anahtar Kavram

Unit digit calculation using cyclicity patterns and modular arithmetic for multi-term exponential expressions.
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Konular

Unit Digit and Cyclicity

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