Exercise on Highest power

by shivamtripathi22 | Aug 29, 2021 | Uncategorized | 0 comments

1. if a! + b! + c! + d! + e! +f! + g! + h! is a two digit number. Find the maximum value of a.

2. If unit digit of x! + y! + z! is 9 then find the value of x!.y!.z!.

3. Find the unit digit of 1! + 2! + 3! + 4! + 5! +6! + ………… +15!

4. If N = p! + q! + r! is a two digit prime number then how many such value of n exists

5. If product of factorial of n consecutive integer is a two digit number then find the maximum value of n.

6. If sum of factorial of n consecutive natural nmbers ia a four digit number then find the maximum value of n.

7. Find highest power of 12 in 1212!

8. Find the highest power of 20 in 10!×20!×30!×40!×50!×60!×70!×80!×90!×100!×110!×120!

9. If N is product of factorial of four consecutive positive integers. Then which of the following is/are correct about N.

(i) if N is a three digit number then there exists two such value of N.
(ii)if N is a Five digit number then there exists only one such value of N
(iii) N may be a four digit number

10. Find the highest power of 10 in 10! + 20! + 30! + 40! + 50! + 60! + 70! + 80! + 90! + 100! + 110! + 120!

11. Find the first non zero digit (FNZD) in 100!

12.Find the first two non zero digit (FNZD) in 100!

13. Find the first non zero digit (FNZD) in 2000!

14. Find the first two non-zero digit(FTNZD) in 2000!

15. Find the first non-zero digit (FNZD) in 20! + 30! +40! + 50! + 60!

16. Find the first non-zero digit (FNZD) in 20! × 30! × 40! × 50! × 60!

17. If the highest power of 10 in N! is 18 then what could be the highest power of 10 in (N+1)!

18. If the highest power of 8 in N! is 19 then find highest power of 8 in (N+1)!

19. A three digit number xyz is such that x + y + z = 26. Then find the sum of all possible values of x!.y!.z!.

20. Find the first two non-zero digit (FTNZD) of $\frac{{555!}}{{444!}}$
(i). 12
(ii). 72
(iii). 96
(iv). 38

21. Find the first two non-zero digit of (555!).(444!).

22. Find the number of trailing zeros in (5²⁰)!.

23. Find the number of trailing zeros in 1!.2!.3!.4!.5!………..148!.149!.150!.

24. If S is the sum of factorials of all the prime numbers less than 150. Then what will be the last three digit of S.

25. If x(y!) is completely divisible by 5¹⁵ & x is a single digit number. then find the minimum & maximum value of y.

26. Let p be a two digit prime number greater than 60 & k be a number such that k= $\frac{{(400!)(300!)}}{{500!}}$ . Given that exponent of p in k is 2. then how many such value of p exists.

27. Find the highest power of 12 in 10² × 11² ×12² × 13² ×……….× 30² × 31².

28. Find the number of trailing zeros at the end of 40×41×42×43×………till 40 terms.

29. Find the highest power of 1000 in 1000×1001×1002×1003×………..till 1000 terms.

30. Find the first two non zero digit i.e. FTNZD in 100×101×102×………..till 100 terms.
(i) 34
(ii) 39
(iii) 24
(iv) 92

31. k is the product of first 100 multiples of 15. Find the number of trailing zeros in k.

32. Given that
k = 1! + 2! + 3! + 4! +5! + ……….. + 120!. Then which one of the following is true
(i) k is odd
(ii) k is even
(iii) k is even or odd, can not be determined
(iv) (k+3) is even.

33. Find the smallest perfect square number divisible by 11!.

34. Consider a number x.(x + 1).(x + 2).(x + 3)………till x terms. If above number is completely divisible by all two digit prime numbers then what is the minimum value of x.

35. Find the number of trailing zeros in (1! + 2! + 3! + 4! +5!)(6! + 7! + 8! + 9! + 10!)…………(96! + 97! + 98! + 99! + 100!).

36. An expression is given below:
N = 1! + 2! + 3! + 4! + 5! + ………. + 99! + 100!
(i) Find the unit digit of N
(ii) Find the last two digit of N
(iii) What will be the remainder when N is divided by 7.

37. If p = ${\left( {x!} \right)^{{{\left( {y!} \right)}^{\left( {z!} \right)}}}}$ & p is a single digit number then find the possible value of p.

38. If p = 10! + 20! + 30! + 40! + ………… + 150!. then find the number of trailing zeros in pᵖ.

39. If p = x¹⁰⁰⁰(1000!) & p is divisible by 10¹⁰⁰⁰. then what would be the minimum number of zeros at the end of p.

40. If k! is not divisible by 1155 then find the maximum value of k
(i) 11
(ii) 17
(iii) 10
(iv) 7

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