Number Theory4 questions

Number Theory — JEE Maths Practice Questions & Solutions

4 questions on Number Theory with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hard
How many of the numbers 12,22,32,…,199921^2, 2^2, 3^2, \ldots, 1999^2 have an odd digit in the tens place?
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hard
The numbers in the sequence 101, 104, 109, 116, …101,\,104,\,109,\,116,\,\ldots are of the form an=100+n2a_n=100+n^2, where n=1,2,3,…n=1,2,3,\ldots. For each nn, let dnd_n be the greatest common divisor of ana_n and an+1a_{n+1}. Let kk be the maximum value of dnd_n as nn ranges over the positive integers. Find the value of k−120\dfrac{k-1}{20}.
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medium
The function ff is defined on the set of integers and satisfies
f(n)={n−3,if n≥1000,f(n+5),if n<1000.f(n)=\begin{cases}n-3, & \text{if } n\ge1000,\\[2pt] f(n+5), & \text{if } n<1000.\end{cases}
 Find the sum of the prime factors of f(84)f(84).
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medium
Let N=22225555+55552222N=2222^{5555}+5555^{2222}. Let rr be the remainder when NN is divided by 462462. Find the sum of the distinct prime factors of rr.
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