Basic Numeracy Alıştırma Soruları — State PSC Exam

Soru 1Soru →

In a sequence of arithmetic operations, a candidate calculates a value of $-82$ . If this integer is then divided by $9$ , determine the least non-negative remainder.

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Cevap: 8

Cevap

8

Applying the division algorithm,

-82

can be expressed in the form

dq + r

as

9 \times (-10) + 8

. Since the remainder

r

must be greater than or equal to

0

and strictly less than the divisor

9

, the least non-negative remainder is

8

.

Adım Adım Çözüm

1

Identify the dividend and the divisor.

Dividend is

-82

, divisor is

9

.

To apply the formal mathematical definition of division with remainders.

2

Find the largest integer multiple of

9

that is less than or equal to

-82

.

The correct multiple is

-90

(since

-90 \leq -82

).

The remainder must be a strictly non-negative value, meaning the multiple we subtract from the dividend must not exceed the dividend.

3

Calculate the difference between the dividend and this multiple to find the remainder.

-82 - (-90) = 8

.

This difference represents the least non-negative remainder required by the problem.

Anahtar Kavram

Remainder Theorem and Negative Dividends

Soru 2Soru →

Determine the positive remainder when the value of $36^{43}$ is divided by $37$ .

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Cevap: 36

Cevap

36

By the properties of modular arithmetic,

36

is congruent to

-1

modulo

37

. When we raise both sides to the power of

43

, we obtain

36^{43} \equiv (-1)^{43} \pmod{37}

. Since an odd power of

-1

is still

-1

, the expression leaves a remainder of

-1

. To find the standard positive remainder, we simply add the divisor to this result:

-1 + 37 = 36

.

Adım Adım Çözüm

1

Express the base as a negative remainder modulo the divisor.

When

36

is divided by

37

, the remainder can be expressed as

-1

because

36 = 37 \times 1 - 1

. Therefore, we can state that

36 \equiv -1 \pmod{37}

.

Using a negative remainder with an absolute value of

1

makes calculating large exponentiations trivial.

2

Apply the exponent to the negative remainder and convert back to a positive value.

Calculate

(-1)^{43}

. Since

43

is an odd number,

(-1)^{43} = -1

. To find the valid positive remainder, add the divisor:

-1 + 37 = 36

.

The rules of modular arithmetic allow us to raise the remainder to the given power. However, the final answer must always be a non-negative integer strictly less than the divisor.

Anahtar Kavram

Negative Remainders for Large Powers

Alternatif Yöntem

You can use Fermat's Little Theorem, which states

a^{p-1} \equiv 1 \pmod p

for any prime

p

(as long as

a

is not divisible by

p

). Here,

p = 37

, so

36^{36} \equiv 1 \pmod{37}

. We can split the expression:

36^{43} = 36^{36} \times 36^7 \equiv 1 \times 36^7 \pmod{37}

. Using the negative remainder trick for the remaining part gives

(-1)^7 = -1

. Adding

37

yields the positive remainder of

36

.

Tahmini Süre:30s

Soru 3Soru →

What is the unit digit of the expression $7^{43} + 9^{27}$ ?

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Cevap: 2

Cevap

The unit digit of the given expression

7^{43} + 9^{27}

is 2.

The unit digit of

7^{43}

is determined by dividing 43 by 4, giving a remainder of 3, so

7^3

ends in 3. The unit digit of

9^{27}

is determined by taking 9 to an odd power, which ends in 9. Adding

3 + 9 = 12

gives a final unit digit of 2.

Adım Adım Çözüm

1

Determine the unit digit of

7^{43}

using cyclicity rules.

The unit digit of

7^{43}

is 3.

The base 7 follows a 4-step cyclicity pattern (7, 9, 3, 1). Dividing exponent 43 by 4 leaves remainder 3, giving

7^3 \pmod{10} = 3

.

2

Determine the unit digit of

9^{27}

using cyclicity rules.

The unit digit of

9^{27}

is 9.

The base 9 follows a 2-step cyclicity pattern (9 for odd powers, 1 for even powers). Since 27 is odd, the unit digit is 9.

3

Combine the resulting unit digits.

The unit digit of the sum is 2.

Summing the individual unit digits gives

3 + 9 = 12

, whose unit digit is 2.

Anahtar Kavram

Unit Digit and Cyclicity

Soru 4Soru →

What is the numerical value of the following mathematical expression evaluated using the BODMAS rule?

12.5\% \text{ of } 480 - \left[ 3 \frac{3}{4} \div \left\{ 1.25 + \frac{1}{2} \text{ of } \left( 3.6 - \overline{1.8 - 0.7} \right) \right\} \right] \times \frac{8}{15}

Cevabı ve açıklamayı göster

Cevap: 59.2

Cevap

The evaluated numerical value of the expression is 59.2.

Following strict BODMAS hierarchy—evaluating the vinculum (1.1), parentheses (2.5), 'of' operation (1.25), curly brackets (2.5), square bracket division (1.5), multiplication (0.8), percentage term (60), and final subtraction—gives 60 - 0.8 = 59.2.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum bar.

1.8 - 0.7 = 1.1

The vinculum has the highest priority and acts as an implicit innermost bracket.

2

Evaluate the terms inside the round brackets (parentheses).

3.6 - 1.1 = 2.5

Round brackets must be resolved after the vinculum.

3

Evaluate the 'of' operation inside the curly brackets.

(1/2) * 2.5 = 1.25

'Of' precedes addition and multiplication in the standard BODMAS order.

4

Complete the addition within the curly brackets.

1.25 + 1.25 = 2.5 = 5/2

Brackets must be fully simplified from inside out.

5

Perform division inside the square brackets.

(15/4) / (5/2) = (15/4) * (2/5) = 3/2 = 1.5

Square brackets are evaluated after inner curly brackets.

6

Multiply the square bracket result by 8/15.

1.5 * (8/15) = (3/2) * (8/15) = 4/5 = 0.8

Multiplication takes precedence over final subtraction.

7

Calculate the initial percentage 'of' term.

12.5% of 480 = 0.125 * 480 = 60

'Of' operation on percentage takes precedence before subtraction.

8

Subtract the bracketed product from the initial term.

60 - 0.8 = 59.2

Subtraction is performed last according to BODMAS.

Anahtar Kavram

BODMAS Order of Operations with Vinculum and Nested Brackets

Tahmini Süre:2m 0s

Soru 5Soru →

What is the unit digit of the expression $S = (1! + 2! + 3! + \dots + 99!) + 3^{99}$ ?

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Cevap: 0

Cevap

The unit digit of the given expression is 0.

The unit digit of a sum is determined by the sum of the unit digits of its individual terms. For the factorial sum

(1! + 2! + 3! + \dots + 99!)

, terms from

5!

onward end in 0 because

5! = 120

. Summing the first four terms yields

1 + 2 + 6 + 24 = 33

, contributing a unit digit of 3. For

3^{99}

, the cyclicity sequence of base 3 is 3, 9, 7, 1 (length 4). Dividing 99 by 4 leaves a remainder of 3, corresponding to

3^3 = 27

, which contributes a unit digit of 7. Summing the two unit digits yields

3 + 7 = 10

, giving a final unit digit of 0.

Adım Adım Çözüm

1

Evaluate the unit digit contribution of the factorial terms

Unit digit of

(1! + 2! + 3! + \dots + 99!)

is 3

Since

n!

ends in 0 for every

n \ge 5

, only

1! + 2! + 3! + 4! = 33

affects the unit digit.

2

Find the unit digit of

3^{99}

using pattern cyclicity

Unit digit of

3^{99}

is 7

The cyclicity of base 3 is 4. Since

99 \pmod 4 = 3

, the unit digit corresponds to

3^3 = 27

.

3

Combine the unit digits of both parts

Unit digit of

S

is 0

Adding the unit digits gives

3 + 7 = 10

, making the final unit digit 0.

Anahtar Kavram

Unit digit evaluation using factorial terminal zeros and exponential cyclicity

Soru 6Soru →

Evaluate the following numerical expression using the correct order of operations (BODMAS):

40 - 4 \times (6 - 2) \div 2 + 5

What is the final result?

37

27

77

17

Cevabı ve açıklamayı göster

Cevap: 37

Cevap

The final result of the expression is 37.

Following the BODMAS rule: first simplify the bracket

(6 - 2) = 4

. Next, execute multiplication and division from left to right:

4 \times 4 = 16

and

16 \div 2 = 8

. The expression reduces to

40 - 8 + 5

. Finally, carry out subtraction and addition from left to right:

40 - 8 = 32

, and

32 + 5 = 37

.

Adım Adım Çözüm

1

Evaluate the expression inside the brackets

6 - 2 = 4

According to BODMAS, operations inside brackets (B) take highest priority.

2

Perform multiplication and division from left to right

4 \times 4 = 16

, then

16 \div 2 = 8

Multiplication and Division (DM) carry equal priority and are evaluated in order from left to right.

3

Perform addition and subtraction from left to right

40 - 8 + 5 = 32 + 5 = 37

Addition and Subtraction (AS) carry equal priority and are evaluated sequentially from left to right.

Anahtar Kavram

BODMAS Order of Operations

Soru 7Soru →

Let $N$ be the smallest positive integer that leaves a remainder of $3$ when divided by $11$ , a remainder of $5$ when divided by $13$ , and is exactly divisible by $7$ . What is the remainder when $N^4 + N^2 + 1$ is divided by $17$ ?

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Cevap: 3

Cevap

The remainder when

N^4 + N^2 + 1

is divided by

17

is 3.

Using the Chinese Remainder Theorem, the smallest integer meeting all divisibility constraints is determined to be

N = 707

. Reducing

707

modulo

17

gives

10

. Substituting

10

into the expression

N^4 + N^2 + 1

modulo

17

yields

(10^4 + 10^2 + 1) \pmod{17} = (4 + 15 + 1) \pmod{17} = 20 \pmod{17} = 3

.

Adım Adım Çözüm

1

Express the given conditions as a system of linear congruences

N \equiv 3 \pmod{11}

,

N \equiv 5 \pmod{13}

, and

N \equiv 0 \pmod{7}

.

Translates the problem statement into standard modular arithmetic notation.

2

Solve the simultaneous congruences for modulo 11 and modulo 13

Let

N = 11k + 3

. Substituting into

11k + 3 \equiv 5 \pmod{13}

gives

11k \equiv 2 \pmod{13} \implies -2k \equiv 2 \pmod{13} \implies k \equiv 12 \pmod{13}

. Thus,

N = 11(13m + 12) + 3 = 143m + 135

.

Combines two modular constraints into a single general expression using the Chinese Remainder Theorem principle.

3

Apply the divisibility condition for 7 to determine the smallest value of

N

143m + 135 \equiv 0 \pmod{7} \implies 3m + 2 \equiv 0 \pmod{7} \implies 3m \equiv 5 \pmod{7} \implies m \equiv 4 \pmod{7}

. For the smallest positive integer, set

m = 4

, so

N = 143(4) + 135 = 707

.

Finds the specific smallest integer

N

that satisfies all three divisor requirements.

4

Reduce

N

modulo 17

707 = 17 \times 41 + 10 \implies N \equiv 10 \pmod{17}

.

Simplifies the base before evaluating higher powers.

5

Calculate the polynomial expression

(N^4 + N^2 + 1) \pmod{17}

N^2 \equiv 10^2 = 100 \equiv 15 \equiv -2 \pmod{17}

, and

N^4 \equiv (-2)^2 = 4 \pmod{17}

. Therefore,

N^4 + N^2 + 1 \equiv 4 + 15 + 1 = 20 \equiv 3 \pmod{17}

.

Uses modular exponentiation properties to efficiently evaluate the final remainder.

Anahtar Kavram

Chinese Remainder Theorem and Modular Exponentiation

Soru 8Soru →

Determine the unit digit of the composite exponential expression $E = 17^{18^{19}} + 18^{19^{20}} - 19^{20^{21}}$ .

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Cevap: 8

Cevap

8

The unit digit of

17^{18^{19}}

is 1 because

18^{19}

is divisible by 4, which corresponds to the 4th power

7^4 \equiv 1 \pmod{10}

. The unit digit of

18^{19^{20}}

is 8 because

19^{20} \equiv 1 \pmod 4

, giving

8^1 = 8

. The unit digit of

19^{20^{21}}

is 1 because

20^{21}

is an even exponent, giving

9^{\text{even}} \equiv 1 \pmod{10}

. Combining these gives

1 + 8 - 1 = 8

.

Adım Adım Çözüm

1

Find the unit digit of

17^{18^{19}}

The unit digit is 1

Base 7 has cyclicity 4. Exponent

18^{19} = 2^{19} \equiv 0 \pmod 4

. When the remainder modulo 4 is 0, we take the 4th power in the cycle:

7^4 \equiv 1 \pmod{10}

.

2

Find the unit digit of

18^{19^{20}}

The unit digit is 8

Base 8 has cyclicity 4. Exponent

19^{20} \equiv (-1)^{20} \equiv 1 \pmod 4

. The 1st power in the cycle gives

8^1 = 8

.

3

Find the unit digit of

19^{20^{21}}

The unit digit is 1

Base 9 has cyclicity 2. The exponent

20^{21}

is even, so

9^{\text{even}} \equiv 1 \pmod{10}

.

4

Evaluate the combined expression

The final unit digit is 8

Combine the unit digits of the individual terms:

1 + 8 - 1 = 8

.

Anahtar Kavram

Cyclicity of numbers and modular arithmetic for tower exponents

Soru 9Soru →

What is the unit digit of the expression $2^{52} + 5^{17}$ ?

1

6

7

3

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The unit digit of the expression

2^{52} + 5^{17}

is 1.

For

2^{52}

, dividing 52 by 4 leaves a remainder of 0. Since the cyclicity of 2 is 4, a remainder of 0 means using

2^4 = 16

, which has a unit digit of 6. For

5^{17}

, any power of 5 ends in 5. The total unit digit is the unit digit of

6 + 5 = 11

, which is 1.

Adım Adım Çözüm

1

Find the unit digit of

2^{52}

using cyclicity.

The cyclicity of 2 is 4 (pattern: 2, 4, 8, 6). Dividing the exponent 52 by 4 yields

52 = 4 \times 13 + 0

. A remainder of 0 corresponds to the 4th power in the cycle.

2^4 = 16

, so the unit digit of

2^{52}

is 6.

When the exponent is exactly divisible by the cyclicity length, the unit digit corresponds to the power equal to the cyclicity length.

2

Find the unit digit of

5^{17}

.

The cyclicity of 5 is 1, meaning any positive integer power of 5 ends in 5. Thus, the unit digit of

5^{17}

is 5.

Base numbers ending in 5 always yield a unit digit of 5 for any positive integer exponent.

3

Add the individual unit digits to get the final unit digit.

Unit digit of

(2^{52} + 5^{17}) = \text{Unit digit of }(6 + 5) = \text{Unit digit of }11 = 1

.

The unit digit of a sum is equal to the unit digit of the sum of the individual unit digits.

Anahtar Kavram

Unit Digit and Cyclicity

Tahmini Süre:45s

Soru 10Soru →

What is the simplified value of the following mathematical expression obtained by applying the BODMAS rule?

96 \div \left[ 4 + \left\{ 25 - \left( 12 - \overline{8 - 3} \right) \times 3 \right\} \right]

12

28

4

16

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

Following the standard BODMAS precedence rule: first evaluate the vinculum

\overline{8-3} = 5

, then the round brackets

(12-5) = 7

, then multiplication

7 \times 3 = 21

, then curly brackets

25 - 21 = 4

, then square brackets

4 + 4 = 8

, and finally division

96 \div 8 = 12

.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

\overline{8 - 3} = 5

The vinculum takes precedence over other bracket operations.

2

Evaluate the operation inside the innermost round brackets

12 - 5 = 7

Resolve the expression within the round brackets next.

3

Perform multiplication inside the curly brackets

7 \times 3 = 21

Multiplication precedes subtraction within the curly brackets according to BODMAS.

4

Complete the subtraction inside the curly brackets

25 - 21 = 4

Evaluate the remaining subtraction inside the curly brackets.

5

Evaluate the addition inside the outer square brackets

4 + 4 = 8

Combine the terms within the square brackets to obtain the final divisor.

6

Perform the final division

96 \div 8 = 12

Divide the initial integer by the evaluated square bracket result.

Anahtar Kavram

Simplification and BODMAS Rule with Vinculum

Soru 11Soru →

What is the unit digit of the expression $E = (842^{324} \times 617^{153}) + 358^{217}$ ?

Cevabı ve açıklamayı göster

Cevap: 0

Cevap

The unit digit of the expression

E = (842^{324} \times 617^{153}) + 358^{217}

is 0.

For

842^{324}

, exponent

324 \pmod 4 = 0

corresponds to the 4th power position in cyclicity (

2^4 = 16 \rightarrow 6

). For

617^{153}

,

153 \pmod 4 = 1

corresponds to

7^1 = 7

. The product of these terms ends in

6 \times 7 = 42 \rightarrow 2

. For

358^{217}

,

217 \pmod 4 = 1

corresponds to

8^1 = 8

. Adding the product's unit digit (2) and

358^{217}

's unit digit (8) gives

2 + 8 = 10 \rightarrow 0

.

Adım Adım Çözüm

1

Find the unit digit of

842^{324}

Unit digit is 6

The unit digit of base 842 is 2, which repeats unit digits in a cycle of 4:

(2, 4, 8, 6)

. Dividing exponent 324 by 4 yields remainder 0, indicating the 4th position in the cycle,

2^4 = 16 \rightarrow 6

.

2

Find the unit digit of

617^{153}

Unit digit is 7

The unit digit of base 617 is 7, which repeats in a cycle of 4:

(7, 9, 3, 1)

. Dividing exponent 153 by 4 yields remainder 1, indicating the 1st position in the cycle,

7^1 = 7

.

3

Calculate the unit digit of the product

(842^{324} \times 617^{153})

Unit digit is 2

Multiplying the unit digits of the two terms gives

6 \times 7 = 42

, which has a unit digit of 2.

4

Find the unit digit of

358^{217}

Unit digit is 8

The unit digit of base 358 is 8, which repeats in a cycle of 4:

(8, 4, 2, 6)

. Dividing exponent 217 by 4 yields remainder 1, indicating the 1st position in the cycle,

8^1 = 8

.

5

Add the unit digit of the product to the unit digit of

358^{217}

Final unit digit is 0

Adding the unit digit of the product (2) and the unit digit of the final term (8) gives

2 + 8 = 10

, which ends in 0.

Anahtar Kavram

Unit Digit Computation using Exponents and Cyclicity

Soru 12Soru →

What is the simplified numerical value of the following mathematical expression when evaluated using the standard BODMAS rule?

240 \div \left[ 2 \frac{1}{2} \times \left\{ 14 - \left( 3 \frac{1}{2} + 25\% \text{ of } \left( 14 - \overline{1.8 + 2.2} \right) \right) \right\} \right]

12

16

768

8

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

Following the strict hierarchical order of operations (BODMAS: Vinculum

\rightarrow

Brackets

\rightarrow

Orders

\rightarrow

Division/Multiplication

\rightarrow

Addition/Subtraction), the innermost vinculum

\overline{1.8 + 2.2}

evaluates to

4

. Then

14 - 4 = 10

, and

25\%

of

10

equals

2.5

. Adding this to

3 \frac{1}{2} (3.5)

yields

6

. Inside the curly brackets,

14 - 6 = 8

. Inside the square brackets,

2 \frac{1}{2} (2.5) \times 8 = 20

. Finally,

240 \div 20 = 12

.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

\overline{1.8 + 2.2} = 4

The vinculum acts as the innermost grouping symbol and must be simplified first.

2

Simplify the innermost parenthetical difference

14 - 4 = 10

Subtract the vinculum result from 14 within the round brackets.

3

Calculate the percentage of the result from Step 2

25\% \text{ of } 10 = \frac{25}{100} \times 10 = 2.5

'Of' represents multiplication in percentage evaluation.

4

Complete the evaluation inside the round brackets

3 \frac{1}{2} + 2.5 = 3.5 + 2.5 = 6

Convert the mixed fraction

3 \frac{1}{2}

to decimal

3.5

and sum with

2.5

.

5

Evaluate the expression inside the curly brackets

14 - 6 = 8

Subtract the round bracket total from 14.

6

Simplify the expression inside the square brackets

2 \frac{1}{2} \times 8 = 2.5 \times 8 = 20

Multiply the mixed fraction

2 \frac{1}{2}

by the curly bracket result.

7

Perform the final division outside the brackets

240 \div 20 = 12

Divide 240 by the completely simplified value of the square brackets.

Anahtar Kavram

BODMAS Rule with Vinculum and Multi-level Brackets

Soru 13Soru →

What is the remainder when the expression $15^{100} + 19^{100}$ is divided by $17$ ?

15

2

1

3

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The remainder when

15^{100} + 19^{100}

is divided by

17

is

15

.

Reducing the bases modulo 17 gives

15 \equiv -2

and

19 \equiv 2

. Since

100

is even,

(-2)^{100} + 2^{100} = 2 \cdot 2^{100} = 2^{101}

. By Fermat's Little Theorem (

2^{16} \equiv 1 \pmod{17}

),

2^{101} \equiv 2^5 = 32 \equiv 15 \pmod{17}

. Thus, the correct remainder is 15.

Adım Adım Çözüm

1

Reduce the base numbers modulo 17

15 \equiv -2 \pmod{17}

and

19 \equiv 2 \pmod{17}

Using negative remainders simplifies power calculations.

2

Rewrite the expression using modular congruence

15^{100} + 19^{100} \equiv (-2)^{100} + 2^{100} = 2^{100} + 2^{100} = 2 \times 2^{100} = 2^{101} \pmod{17}

Since the exponent

100

is even,

(-2)^{100} = 2^{100}

.

3

Apply Fermat's Little Theorem to simplify the exponent

2^{16} \equiv 1 \pmod{17}

17 is prime and

\gcd(2, 17) = 1

.

4

Divide the exponent 101 by 16 to find the remaining power

101 = 16 \times 6 + 5 \implies 2^{101} \equiv (2^{16})^6 \times 2^5 \equiv 1^6 \times 32 \equiv 32 \pmod{17}

Any integer power of 1 is 1 under modular arithmetic.

5

Compute the final positive remainder

32 = 17 \times 1 + 15 \implies 32 \equiv 15 \pmod{17}

(or

-2 + 17 = 15

)

Remainders must be non-negative integers strictly less than the divisor.

Anahtar Kavram

Fermat's Little Theorem and Negative Remainders in Modular Arithmetic

Soru 14Soru →

What is the final value of the expression $36 \div 4 \times 3 + (15 - 7) - 10$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The correct answer is 25.

Applying the BODMAS order of operations strictly: first simplify the bracket

(15 - 7) = 8

. Next, evaluate multiplication and division from left to right:

36 \div 4 = 9

and

9 \times 3 = 27

. Finally, evaluate addition and subtraction left to right:

27 + 8 - 10 = 25

.

Adım Adım Çözüm

1

Evaluate bracket expression

15 - 7 = 8

Brackets have the highest precedence in the BODMAS rule.

2

Perform division and multiplication from left to right

36 ÷ 4 = 9, then 9 × 3 = 27

Division and multiplication carry equal priority and are evaluated in order from left to right.

3

Perform addition and subtraction from left to right

27 + 8 - 10 = 25

Addition and subtraction are performed sequentially from left to right.

Anahtar Kavram

BODMAS Order of Operations

Tahmini Süre:45s

Soru 15Soru →

In basic arithmetic, the unit digit of a power of an integer repeats in a periodic pattern known as cyclicity. What is the unit digit of $3^{45}$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The unit digit of

3^{45}

is 3.

The base number 3 has a cyclicity pattern of length 4: (3, 9, 7, 1). When the exponent 45 is divided by 4, the quotient is 11 and the remainder is 1. A remainder of 1 indicates the unit digit is equal to the first element of the cyclic pattern, which is 3.

Adım Adım Çözüm

1

Identify the unit digit cyclicity pattern for base 3.

The unit digits repeat every 4 powers in the pattern (3, 9, 7, 1).

Calculating sequential powers of 3 yields last digits of 3, 9, 7, 1, 3, 9, 7, 1, establishing a cycle length of 4.

2

Divide the exponent by the cycle period of 4.

45 \div 4 = 11

with a remainder of

1

.

The remainder indicates the exact position within the repeating cycle.

3

Find the unit digit using the remainder.

Since the remainder is 1, the unit digit is

3^1 = 3

.

The first number in the cyclicity sequence (3, 9, 7, 1) is 3.

Anahtar Kavram

Unit Digit Cyclicity

Tahmini Süre:45s

Soru 16Soru →

Evaluate the following mathematical expression using the standard BODMAS rule:

40\% \text{ of } 150 + \left[ 72 \div \left\{ 18 - \left( 2\frac{1}{2} \times 4 + \overline{11 - 5} \right) \right\} \right]

What is the final simplified numerical value?

Cevabı ve açıklamayı göster

Cevap: 96

Cevap

The simplified numerical value of the given expression is 96.

Applying the standard BODMAS rule: first evaluate the vinculum

\overline{11 - 5} = 6

. Next, evaluate inside the round brackets:

2.5 \times 4 + 6 = 16

. Then simplify inside the curly braces:

18 - 16 = 2

. Following that, perform division inside the square brackets:

72 \div 2 = 36

. Separately, evaluate the percentage term:

40\% \text{ of } 150 = 60

. Summing these two terms gives

60 + 36 = 96

.

Adım Adım Çözüm

1

Evaluate the bar (vinculum) expression.

\overline{11 - 5} = 6

The vinculum (bar) has highest priority within bracket operations.

2

Simplify the terms within the round parentheses.

2\frac{1}{2} \times 4 + 6 = 10 + 6 = 16

Perform multiplication before addition inside parentheses.

3

Evaluate the expression inside the curly braces.

18 - 16 = 2

Subtract the result of the round brackets from 18.

4

Perform the division inside the square brackets.

72 \div 2 = 36

Divide 72 by the value obtained from curly braces.

5

Calculate the percentage component of the expression.

40\% \text{ of } 150 = \frac{40}{100} \times 150 = 60

The word 'of' indicates multiplication applied to the percentage value.

6

Perform the final addition.

60 + 36 = 96

Add the result of the percentage calculation to the result of the bracket evaluation.

Anahtar Kavram

Order of Operations (BODMAS Rule) with Vinculum, Nested Brackets, and Percentages

Tahmini Süre:1m 30s

Soru 17Soru →

Evaluate the following mathematical expression by strictly applying the BODMAS rule:

150\% \text{ of } 48 - \left[ 14.5 - \left\{ 4 \frac{1}{2} \times \left( 12.8 \div \overline{2.5 + 1.5} - 1 \frac{3}{5} \right) + 2.8 \right\} \right] \div 0.15

What is the simplified numerical value?

Cevabı ve açıklamayı göster

Cevap: 42

Cevap

The simplified numerical value of the given mathematical expression is 42.

Following the strict BODMAS order of operations: Vinculum -> Round brackets -> Curly brackets -> Square brackets -> Division -> 'Of' -> Subtraction gives the exact result 42.

Adım Adım Çözüm

1

Evaluate the term under the vinculum (bar)

2.5 + 1.5 = 4

The vinculum has highest priority among grouping symbols.

2

Evaluate operations inside the round brackets (parentheses)

12.8 / 4 - 1.6 = 3.2 - 1.6 = 1.6

Division inside parentheses precedes subtraction, and 1 3/5 converts to decimal 1.6.

3

Evaluate operations inside the curly brackets

4.5 * 1.6 + 2.8 = 7.2 + 2.8 = 10

Multiplication precedes addition inside curly brackets, with 4 1/2 converting to 4.5.

4

Evaluate operations inside the square brackets

14.5 - 10 = 4.5

Subtract the evaluated inner expression from 14.5.

5

Perform the division following the bracket result

4.5 / 0.15 = 30

Division operation takes precedence over final subtraction.

6

Calculate the percentage 'of' expression

150% of 48 = (150 / 100) * 48 = 72

'Of' operation is evaluated before basic addition and subtraction.

7

Compute the final subtraction

72 - 30 = 42

Subtract the result of the bracketed division term from the percentage evaluation.

Anahtar Kavram

Order of Operations (BODMAS / PEMDAS) with Vinculum and Nested Brackets

Soru 18Soru →

Find the value of the given mathematical expression by applying the standard order of operations (BODMAS):

25 - [12 + \{8 \div 2 \times (5 - \overline{3 + 1})\}]

9

15

17

1

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

Following the BODMAS rule: evaluate the vinculum

\overline{3 + 1} = 4

, then round brackets

(5 - 4) = 1

, then curly brackets

\{8 \div 2 \times 1\} = 4

, then square brackets

[12 + 4] = 16

, and finally

25 - 16 = 9

.

Adım Adım Çözüm

1

Evaluate the expression under the vinculum (bar)

3 + 1 = 4

The vinculum has the highest precedence among brackets (VBODMAS).

2

Evaluate the innermost round bracket

5 - 4 = 1

Round brackets are evaluated after resolving the vinculum.

3

Evaluate division and multiplication inside curly brackets from left to right

8 ÷ 2 × 1 = 4 × 1 = 4

Division and multiplication have equal precedence and are resolved from left to right.

4

Evaluate addition inside square brackets

12 + 4 = 16

Square brackets are resolved next.

5

Perform final subtraction outside the brackets

25 - 16 = 9

Outer subtraction is performed as the final step.

Anahtar Kavram

Order of Operations (BODMAS / VBODMAS)

Tahmini Süre:45s

Soru 19Soru →

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

4

9

6

0

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The unit digit of the expression is 4.

Evaluating each term using cyclicity:

432^{316}

ends in 6 (

2^4

),

657^{235}

ends in 3 (

7^3

),

875^{432}

ends in 5 (

5^n

), and

959^{411}

ends in 9 (

9^{\text{odd}}

). Combining these gives

(6 \times 3) + 5 - 9 = 18 + 5 - 9 = 14

, yielding a unit digit of 4.

Adım Adım Çözüm

1

Determine the unit digit of

432^{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:

2^4 = 16

, giving a unit digit of 6.

2

Determine the unit digit of

657^{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 \times 58 + 3

). We take the 3rd power in the cycle:

7^3 = 343

, giving a unit digit of 3.

3

Calculate the unit digit of the product

(432^{316} \times 657^{235})

Unit digit is 8

Multiplying the unit digits of the two terms yields

6 \times 3 = 18

, which has a unit digit of 8.

4

Determine the unit digit of

875^{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 = 13

, so the sum ends in 3.

5

Determine the unit digit of

959^{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,

959^{411}

ends in 9. Subtracting this from the sum's unit digit gives

3 - 9 \equiv 13 - 9 = 4

.

Anahtar Kavram

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

Soru 20Soru →

A positive integer $N$ when successively divided by $4$ , $5$ , and $6$ leaves remainders of $2$ , $3$ , and $4$ , respectively. What is the sum of the remainders obtained when the smallest such number $N$ is successively divided by $6$ , $5$ , and $4$ , in that order?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The sum of the remainders obtained when the smallest such number is successively divided by 6, 5, and 4 is 7.

The smallest positive integer

N

satisfying the given successive division conditions is 94. Successively dividing 94 by 6, 5, and 4 yields quotients of 15, 3, and 0 with remainders 4, 0, and 3, respectively. The sum of these remainders is

4 + 0 + 3 = 7

.

Adım Adım Çözüm

1

Formulate the algebraic equations for successive division and calculate the smallest positive value of

N

.

N = 94

By definition of successive division,

N = 4q_1 + 2

,

q_1 = 5q_2 + 3

, and

q_2 = 6q_3 + 4

. Setting the non-negative integer quotient

q_3 = 0

yields

q_2 = 4

,

q_1 = 23

, and

N = 94

.

2

Perform successive division of 94 by the divisors 6, 5, and 4 in sequence.

The sequence of remainders is

4

,

0

, and

3

.

First stage:

94 \div 6 = 15

remainder

4

. Second stage:

15 \div 5 = 3

remainder

0

. Third stage:

3 \div 4 = 0

remainder

3

.

3

Calculate the sum of the three remainders.

4 + 0 + 3 = 7

Adding the individual remainders obtained from each stage of the reverse order division.

Anahtar Kavram

Successive Division and Remainder Property

Basic Numeracy

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