stevecultt wrote:What is the remainder when p is divided by 10?
(1) p^(11) +‚ 11^p, when divided by 10, leaves remainder 4.
(2) p^3, when divided by 10, leaves remainder 3.
OA D
Statement 1:
We know that: p^11 ‚+ 11^p, when divided by 10, leaves remainder 4
=ƒ> The unit digit of (p^11 ‚+ 11^p) is 4
The unit digit of 11^p = The unit digit of 1^p ƒ =1
=> The unit digit of p^11 = 4 - 1 = 3
Since the unit digit of p^11 is odd, p must be odd.
Thus, possibilities for the unit digit of p are 1, 3, 5, 7 and 9:
- *Unit digit of p is 1: The unit digit of 1^11 ƒ= 1 ≠3 - Does not satisfy
*Unit digit of p is 3: The unit digit of 3^11 = The unit digit of 3^3 ƒ= 7 - Does not satisfy
(The unit digit cycle for exponents of 3 is: 3, 9, 7, 1)
*Unit digit of p is 5: The unit digit of 5^11 ƒ= 5 - Does not satisfy
(The unit digit for exponents of 5 is always 5)
*Unit digit of p is 7: The unit digit of 7^11 = The unit digit of 7^3 = 3 - Satisfies
(The unit digit cycle for exponents of 7 is: 7, 9, 3, 1)
*Unit digit of p is 9: The unit digit of 9^11 = The unit digit of 9^1 = 9 - Does not satisfy
(The unit digit cycle for exponents of 9 is: 9, 1)
Thus, the unit digit of p is 7.
=ƒ> The required remainder is 7. - Sufficient
Statement 2:
We know that: p^3, when divided by 10, leaves remainder 3
ƒ=> The unit digit of p^3 is 3
Since the unit digit of p^3 is odd, p must be an odd number.
Thus, possibilities for the unit digit of p are 1, 3, 5, 7 and 9:
- Unit digit of p is 1: The unit digit of 1^3 ƒ= 1 - Does not satisfy
Unit digit of p is 3: The unit digit of 3^3 =ƒ 7 - Does not satisfy
Unit digit of p is 5: The unit digit of 5^3 =ƒ 5 - Does not satisfy
Unit digit of p is 7: The unit digit of 7^3 ƒ= 3 - Satisfies
Unit digit of p is 9: The unit digit of 9^3 ƒ= 9 - Does not satisfy
Thus, the unit digit of p is 7.
ƒ=> The required remainder is 7. - Sufficient
The correct answer:
D
Hope this helps!
Relevant book:
Manhattan Review GMAT Data Sufficiency Guide
-Jay
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