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DSSSB TGT Number Theory — practice questions

Explore 37 free Number Theory questions in DSSSB TGT Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Number Theory questions and exercises with answers

Showing questions 1–30 of 37.

For any integer $n$, the expression $n^2 - 1$ is always divisible by 8 when $n$ isRead the answer and worked solution →The product of any three consecutive positive integers is always divisible byRead the answer and worked solution →The number of trailing zeros at the end of $50!$ isRead the answer and worked solution →If $\text{HCF}(a, b) = 12$ and $\text{LCM}(a, b) = 180$, then the number of possible ordered pairs $(a, b)$ isRead the answer and worked solution →The largest number that divides 245 and 1029, leaving remainders 5 and 5 respectively, isRead the answer and worked solution →For some integer $m$, every odd integer can be written in the formRead the answer and worked solution →The square of any odd integer, when divided by 8, leaves remainderRead the answer and worked solution →The decimal expansion of $\dfrac{257}{2^3 \cdot 5^4}$ terminates after how many decimal places?Read the answer and worked solution →How many positive integers less than 1000 are divisible by neither 2 nor 5?Read the answer and worked solution →If $p$ is a prime and $p \mid a^2$, then which must be true?Read the answer and worked solution →The unit digit of $7^{2026}$ isRead the answer and worked solution →The smallest positive integer that is divisible by every integer from 1 to 10 isRead the answer and worked solution →If $a = 2^3 \cdot 3^4 \cdot 5^2$ and $b = 2^4 \cdot 3^2 \cdot 7$, then $\dfrac{\text{LCM}(a,b)}{\text{HCF}(a,b)}$ equalsRead the answer and worked solution →Which of the following is an irrational number?Read the answer and worked solution →The remainder when $2^{100}$ is divided by 7 isRead the answer and worked solution →Two positive integers have HCF 9 and LCM 90. If one number is 18, the other isRead the answer and worked solution →The HCF of two consecutive positive integers isRead the answer and worked solution →If $3^x = 4^y = 12^z$ with $x,y,z$ nonzero, then $\dfrac{1}{z}$ equalsRead the answer and worked solution →Complete the definition of slopeRead the answer and worked solution →True or false: slopeRead the answer and worked solution →Zero slope, undefined slope, or neither?Read the answer and worked solution →Match each condition to what it tells youRead the answer and worked solution →Parallel, perpendicular, or neither?Read the answer and worked solution →True or false: parallel and perpendicularRead the answer and worked solution →Order the steps: test three points for collinearityRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Which test applies?Read the answer and worked solution →Flip and pair: condition with conclusionRead the answer and worked solution →

Worked practice questions

For any integer $n$, the expression $n^2 - 1$ is always divisible by 8 when $n$ is

  1. any integer
  2. an even integer
  3. a prime number
  4. an odd integer
Show answer and solution

Correct answer: D. an odd integer

If $n$ is odd, write $n = 2k+1$. Then $n^2 - 1 = 4k(k+1)$. Since $k(k+1)$ is a product of two consecutive integers it is even so $4k(k+1)$ is divisible by 8. For even $n$, $n^2-1$ is odd, so not divisible by 8.

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The product of any three consecutive positive integers is always divisible by

  1. 4
  2. 12
  3. 9
  4. 6
Show answer and solution

Correct answer: D. 6

Among any three consecutive integers, at least one is divisible by 2 and exactly one is divisible by 3. Hence the product is divisible by $2 \times 3 = 6$. It need not be divisible by 4 (e.g. $1\cdot2\cdot3 = 6$).

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The number of trailing zeros at the end of $50!$ is

  1. 12
  2. 13
  3. 11
  4. 10
Show answer and solution

Correct answer: A. 12

Trailing zeros are governed by the power of 5 in $50!$: $\lfloor 50/5 \rfloor + \lfloor 50/25 \rfloor = 10 + 2 = 12$. (Powers of 2 are more plentiful, so 5 is the limiting factor.)

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More Mathematics topics

SequencesBinomial TheoremGeometryProbability & StatisticsSequences and SeriesVectors & 3D GeometryProbabilityCorrelation and Regression