stonecold wrote:
Find the units digit of (556)^17n + (339)^(5m+15n), where m and n are positive integers
(1) 4m+12n = 360
(2) n is the smallest 2-digit positive integer divisible by 5
556 has the units digit of 6 so it will always end in 6. We don't need to worry about the exponent here.
339 has the units digit of 9 so its cyclicity is 9, 1, 9, 1, 9, 1, 9, 1... etc.
If the exponent is odd, 339 will end with 9. If the exponent is even, 339 will end with 1.
We need to find whether (5m+15n) is odd or even.
(1) 4m+12n = 360
m + 3n = 90
There are only two ways in which we can get an even sum.
Odd + Odd = Even or
Even + Even = Even
So m and n are either both odd or both even.
Hence 5m +15n must be even only. So 339^{5m + 15n} ends with a 1.
Hence, (556)^17n + (339)^(5m+15n) will have units digit of 6+1 = 7.
Sufficient
(2) n is the smallest 2-digit positive integer divisible by 5
n must be 10 - even
But we don't know about m. If it is odd, (5m+15n) will be odd. If it is even, (5m+15n) will be even.
Not sufficient.
Answer (A)
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Karishma
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