Practice Questions
Last Two Digits
Find n mod 100. Use cyclicity mod 100 or binomial expansion for numbers ending in 1, 3, 7, 9.
Last Three Digits
Find n mod 1000. Useful for large powers and product problems in placement exams.
Techniques
- Work mod 100 or mod 1000 directly.
- Split the base into tens and units when squaring.
- Use Euler's theorem for advanced cases (φ(100)=40).
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Q1 Medium
Last two digits of 7^4 are:
Explanation7^2=49, 7^4=49^2=2401. Last two digits 01. -
Q2 Easy
Last two digits of 13^2 are:
Explanation13² = 169. Last two digits 69. -
Q3 Medium
Last three digits of 5^10 are:
Explanation5^10 = 9765625. Last three digits 625. -
Q4 Medium
Last two digits of 2^10 are:
Explanation2^10 = 1024. The last two digits are 24. -
Q5 Easy
Last two digits of 76^2 are:
Explanation76² = 5776. Last two digits 76 (pattern for …76). -
Q6 Hard
Last two digits of 3^20 are:
Explanation3^20 = (3^10)^2. 3^5=243, 3^10 last two 49, 49^2=2401 → 01. -
Q7 Easy
Last three digits of 12^3 are:
Explanation12³ = 1728. Last three digits 728.
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Q8 Easy
Last two digits of 19^2 are:
Explanation19² = 361. Last two digits 61. -
Q9 Hard
Last two digits of 4^15 are:
Explanation4^5 = 1024 (ends 24). 24^3 = 13824, so the last two digits are 24. -
Q10 Easy
Last three digits of 7^3 are:
Explanation7³ = 343. -
Q11 Medium
Last two digits of 25^10 are:
ExplanationPowers of 25 end in 25 for n≥2 (25, 625, …). -
Q12 Medium
Last two digits of 11^4 are:
Explanation11^4 = 14641. The last two digits are 41. -
Q13 Easy
Last three digits of 2^8 are:
Explanation2^8 = 256. -
Q14 Easy
Last two digits of 37^2 are:
Explanation37² = 1369. Last two digits 69.
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DS1 Medium
What are the last two digits of 7^n?
Statement In = 4.
Statement IIn is even.
Explanation7^4 = 2401 → last two digits 01. Statement II alone does not fix n. -
DS2 Easy
What are the last three digits of 5^n for n ≥ 3?
Statement In = 10.
Statement IIn is greater than 2.
Explanation5^10 ends in 625. For n ≥ 3, last three digits of 5^n are 125 or 625; n = 10 fixes 625.
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DS3 Medium
What are the last two digits of 13^2?
Statement I13 is prime.
Statement IIThe number is a perfect square.
Explanation13 prime or perfect square alone does not identify 13². Together they still do not fix a unique square beyond context. -
DS4 Easy
What are the last two digits of N?
Statement IN = 76^2.
Statement IIN is a four-digit number.
Explanation76² = 5776 → last two digits 76. Statement II alone does not determine N.