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If a positive integer t is not divisible by 5, how many possible different remainders can t^4 have when it is divided by 5?
A. one
B. two
C. three
D. four
E. five
_________________
A. one
B. two
C. three
D. four
E. five
_________________
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One remained only. Everytime the remainder will be one
Posted from my mobile device
Posted from my mobile device
Given:
* t is a positive integer
* t is not divisible by 5 OR t/5 doesn't equal a whole number
Rephrasing of the question:
How many possible different remainders can t^4 have when it is divided by 5?
OR
Find all the possible remainders of t^4/5 and how many of them are they?
Applicable number properties:
* All multiples of 5 have ones place digits of 5 or 0.
* 1-4 & 6-9 are the set of possible ones place digits not divisible by 5.
Ones place digits for integers (not 5) to the 4th power:
- Ones place digit for 1^4 = 1
- Ones place digit for 2^4 = 6
- Ones place digit for 3^4 = 1
- Ones place digit for 4^4 = 6
- Ones place digit for 6^4 = 6
- Ones place digit for 7^4 = 1
- Ones place digit for 8^4 = 6
- Ones place digit for 9^4 = 1
Thus, all possible remainders for t^4/5 will have a ones place digit of 6 or 1 and because all multiples of 5 end in 5 or 0, we can conclude that all remainders of t^4/5 will have a remainder of 1 because all numbers with a ones digit place of 6 or 1 to be divided by 5 will have 1 as the remainder.
Answer is A, just the one single remainder answer.
* t is a positive integer
* t is not divisible by 5 OR t/5 doesn't equal a whole number
Rephrasing of the question:
How many possible different remainders can t^4 have when it is divided by 5?
OR
Find all the possible remainders of t^4/5 and how many of them are they?
Applicable number properties:
* All multiples of 5 have ones place digits of 5 or 0.
* 1-4 & 6-9 are the set of possible ones place digits not divisible by 5.
Ones place digits for integers (not 5) to the 4th power:
- Ones place digit for 1^4 = 1
- Ones place digit for 2^4 = 6
- Ones place digit for 3^4 = 1
- Ones place digit for 4^4 = 6
- Ones place digit for 6^4 = 6
- Ones place digit for 7^4 = 1
- Ones place digit for 8^4 = 6
- Ones place digit for 9^4 = 1
Thus, all possible remainders for t^4/5 will have a ones place digit of 6 or 1 and because all multiples of 5 end in 5 or 0, we can conclude that all remainders of t^4/5 will have a remainder of 1 because all numbers with a ones digit place of 6 or 1 to be divided by 5 will have 1 as the remainder.
Answer is A, just the one single remainder answer.
If "t" is not divisible by 5 then it can take one of the following forms:-
(5n+1) or (5n+2) or (5n+3) or (5n+4).
Now let's take first case
1. t = 5n+1
So for checking divisibility of (5n+1)^4 by 5, we need to expand this expression. All the terms of its expansion will have some power of 5 except one!
It is:- 4C0 x (5n)^0 x (1)^4 = 1.
So the remainder is "1".
2. For (5n+2)^4:-
That one exceptional term is:- 4C0 x (5n)^0 x (2)^4 = 16.
So here also remainder is "1".
3. For (5n+3)^4:-
That one exceptional term is:- 4C0 x (5n)^0 x (3)^4 = 81
So here also remainder is "1"
And 4. For (5n+4)^4:-
That one exceptional term is:- 4C0 x (5n)^0 x (4)^4 = 256
So here also remainder is "1".
So for all cases the remainder is "1", so there is only one possible remainder.
Posted from my mobile device
(5n+1) or (5n+2) or (5n+3) or (5n+4).
Now let's take first case
1. t = 5n+1
So for checking divisibility of (5n+1)^4 by 5, we need to expand this expression. All the terms of its expansion will have some power of 5 except one!
It is:- 4C0 x (5n)^0 x (1)^4 = 1.
So the remainder is "1".
2. For (5n+2)^4:-
That one exceptional term is:- 4C0 x (5n)^0 x (2)^4 = 16.
So here also remainder is "1".
3. For (5n+3)^4:-
That one exceptional term is:- 4C0 x (5n)^0 x (3)^4 = 81
So here also remainder is "1"
And 4. For (5n+4)^4:-
That one exceptional term is:- 4C0 x (5n)^0 x (4)^4 = 256
So here also remainder is "1".
So for all cases the remainder is "1", so there is only one possible remainder.
Posted from my mobile device
If a positive integer t is not divisible by 5, how many possible different remainders can t^4 have when it is divided by 5?
Remainder of a number by 5 = Remainder of Units' digit of the number by 5
=> We need to find possible value of units' digit of t^4 where t can be any positive integer but not a multiple of 5.
Now, for t possible value of units's digit = 1, 2 , 3, 4, 6, 7, 8, 9 (it cannot be 0 and 5 as then it will become a multiple of 5)
=> Units' digit of t^4 when
=> Possbile units' digits are 1 and 6
And Remainders of both 1 and 6 by 5 is 1
So, Answer will be A
Hope it Helps!
Watch following video to MASTER Remainders by 2, 3, 5, 9, 10 and Binomial Theorem
_________________
Remainder of a number by 5 = Remainder of Units' digit of the number by 5
=> We need to find possible value of units' digit of t^4 where t can be any positive integer but not a multiple of 5.
Now, for t possible value of units's digit = 1, 2 , 3, 4, 6, 7, 8, 9 (it cannot be 0 and 5 as then it will become a multiple of 5)
=> Units' digit of t^4 when
- Units' digit of t = 1 will be = Units' digit of 1^4 = 1
- Units' digit of t = 2 will be = Units' digit of 2^4 = 6
- Units' digit of t = 3 will be = Units' digit of 3^4 = 1
- Units' digit of t = 4 will be = Units' digit of 4^4 = 6
- Units' digit of t = 6 will be = Units' digit of 6^4 = 6
- Units' digit of t = 7 will be = Units' digit of 7^4 = 1
- Units' digit of t = 8 will be = Units' digit of 8^4 = 6
- Units' digit of t = 9 will be = Units' digit of 9^4 = 1
=> Possbile units' digits are 1 and 6
And Remainders of both 1 and 6 by 5 is 1
So, Answer will be A
Hope it Helps!
Watch following video to MASTER Remainders by 2, 3, 5, 9, 10 and Binomial Theorem
_________________
How to Solve :
All Quant Topics Theory || Last Two Digits of Exponents Ending with 1, 2, 3, 4, 5, 6, 7, 8, 9 || Units' Digit Problems || Even and Odd || Divisibility Rules || Positive and Negative || Absolute Value Basics || Inequalities || Absolute Value + Inequality || Functions and Custom Characters || LCM & GCD || Remainders || Factors and Multiples || Arithmetic and Geometric Progression || Co-ordinate Geometry || Circles || Probability: Coin Toss || Work Rate || Statistics || Stem & Leaf Plot || Box And Whisker Plot
GMAT Club Topic Wise Problems
All Quant Topics Theory || Last Two Digits of Exponents Ending with 1, 2, 3, 4, 5, 6, 7, 8, 9 || Units' Digit Problems || Even and Odd || Divisibility Rules || Positive and Negative || Absolute Value Basics || Inequalities || Absolute Value + Inequality || Functions and Custom Characters || LCM & GCD || Remainders || Factors and Multiples || Arithmetic and Geometric Progression || Co-ordinate Geometry || Circles || Probability: Coin Toss || Work Rate || Statistics || Stem & Leaf Plot || Box And Whisker Plot
GMAT Club Topic Wise Problems
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