Iquanta Numbers

1.How many 3-digit numbers less than 500 exist which have 12 factors and only 2 prime factors 2. 161^199 mod 25 find remainder 3. (2^2-1)(3^2-1)....(11^2-1) mod 13 4. Sum of all values of N which satisfies 1<= N <= 200 and N^2 mod 100 = 21. 5. What would be the remainder when 2^110001^11^110000 1is divided by 23. Last two digits of 3^3^3^3^3......(2012 times) ? What would be the remainder when 3^2002 is divided by 17 ? The number 2^1000 is divided by 13. What is the remainder? Last digit of 7^2013 ? Last two digits of 2^2013 ? Remainder when 2^100000 is divided by 17 ? 11^104 + 1 is divided by 17 ? If n is relatively prime to 72 then n¹² mod 72 is 1. Remainder when 29^202 is divided by 13. Remainder when 99^999999 is divided by 23 . Remainder when 29^198 is divided by 20. Remainder when 3^1000000 is divided by 14. a, b, c are digits taking values from 0 to 9. What is the smallest 6-digit number of the form abcabc that has 24 factors? 3333......33333 repeated 2014 times, what is the remainder when this number is divided by 45?

Ans key 1)2 2)87(for last 2 digits, it's the value we get dividing with 100) 3)9 4)3 5)7

6)92,similar process as of second 7)1 8)0 9) is it of the form 72k+1? N can be 73? 10)3 11)22 12)1 13)11 14)abc(1001)=abc*7*11*13.... To make it a 6 digit number we can multiply with 17^2(next possible prime^2) (factors 2*2*2*3) 15)remainder when divided by 9 and 5...got 3 for both... 9a+3,4b+3... Combining them got 48 /45=3

6. Sum of all values of N which satisfies 1<= N <= 200 and N^2 mod 100 = 21. 7. N=1223334444... 301 digits Find remainder when divided by 250 8. 11111......1038 digits mod 53 9. If a three digit number ‘abc’ has 2 factors (where a, b, c are digits), how many factors does the 6-digit number ‘abcabc’ have? 10. What would be remainder when (3^100-2^100 ) is divided by 11 ? 11 What digit does “a” represent, if 35! = 10333147966386144929a66651337523200000000 ? 12. Remainder when 4^4^4 is divided by 32 13. What would be last two digits of (121^47) -(149^124) ? 14. max rem we can get when a 2 digit no is divided by sum of it's digits 15. Find the remainder when 5+55+555+5555+....99 times is divided by 400. 16. N =5^(1! +2! +3! +......+12! +13!) The digits of the number N are added to get another number. Then the digits of the number obtained are added to get yet another number. The process is repeated till a single digit number is obtained. What is that single digit number? (a) 7 (b) 6 (c) 5 (d) 8

17. For which integer n is (2^8 + 2 ^11 + 2^n) a perfect square a)11 b) 12 c) 13 d)14 e) none of these.

18. What would be remainder when 6 +66 + 666 + 6666 +...........+666666....(6 comes 66 times) is divided by 13 ? 19. Remainder when 67676767....(67 repeated 900 times) is divided by 999 ? 20. 5^5555 + 10^5555 + 15^5555 + ... + 5555^5555 mod 100 21. z is product of first 31 natural no. if x=z+1 then no of primes among x+1,x+2,...x+30 is ?? 22. Find the largest number P that can be obtained as the product of two or more positive integers whose sum is 20 1) 625 2) 1024 3) 1458 4) 1728 5) 1998

22. 54355435....200 digits mod 101 = ? 23. 425425425...180 digits mod 27 = ? 24. 345345...302 digits mod 999 = ? 25. 543345543345.....600 digits mod 13 =? 26. 38^33^41mod11? 27. Find the Remainder when 7^7^7 / 13? 28. 161^199 mod 25 29. Find the Remainder when 8^643 / 132? 30. For the number 2450 find 1. The sum and the number of factor divisible by 35 2. The sum and the number of factor divisible by 245.

31. Find n where number of zero's in n! is 11?... Ans: No such number exists. Skip zeroes :5,11,17,23,.. there is no n! with these number of zeroes. 32. Find the Remainder when 7^7^7 / 13 ?

33. Last digit of 1^1! + 2^2! + ... + 81^81! 34. Sum of all values of N which satisfies 1<= N <= 200 and N^2 mod 100 = 21. 35. (2^2-1)(3^2-1)....(11^2-1) mod 13 36. How many natural numbers are factors of at least one of 6^14,10^12 and 15^20? 37.