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25 Jul 2023, 08:00
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A
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49%
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When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?
(1) x + y is a multiple of 9.
(2) x*y divided by 9, the remainder is 5
(1) x + y is a multiple of 9.
(2) x*y divided by 9, the remainder is 5
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25 Jul 2023, 08:28
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Bunuel
x = 9p + m ; 0 <= m < 9
y = 9q + n ; 0 <= n < 9
0 < m + n < 18
1) x + y = 9(p+q) + (m+n)
9(p+q) is a multiple of 9. So, (m+n) must be a multiple of 9. Between 0 and 18, (m + n) must take a minimum value of 9. The sum cannot be zero as m > n.
Sufficient.
(2) x*y divided by 9, the remainder is 5
x * y = (9p + m) * (9q + n)
x * y = 9^2*p*q + 9*p*n + 9*q*m + m * n
9^2*p*q + 9*p*n + 9*q*m is divisible by 9.
Hence,
Remainder m * n/5 = 5
1) m*n = 5
m = 5, and n = 1
2) m*n = 9 + 5 = 14
m = 7 and n = 2
Not sufficient.
IMO A
General Discussion
limuengaoey
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25 Jul 2023, 08:21
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When positive integer x is divided by 9, the remainder is m and when positive integer y is divided by 9, the remainder is n. If m > n, what is the value of m + n ?
x = 9p+ m
y= 9q +n
but m>n hence m cannot be 1 and n cannot be 8
hence m = 2,3,4,...,8
n = 1,2,3,...,7
(1) x + y is a multiple of 9.
multiple of 9 = 9,18,27
suppose x+y = 18
x=15,y=3 then m=6, n=3 hence m+n = 9
suppose x+y = 27
x=14, y = 13 then m=5,n = 4 m+n =9
suppose x=15, n = 12; m=6,n=3 m+n = 9
hence 1 is enough then the answer could be A or D
(2) x*y divided by 9, the remainder is 5
x*y= (9p+m)*(9q+n)=81pq+9pn+9qn+mn from this we know that m*n is 5
so m must be 5 and n must be 1 only then m+n =6 enough
answer is D
x = 9p+ m
y= 9q +n
but m>n hence m cannot be 1 and n cannot be 8
hence m = 2,3,4,...,8
n = 1,2,3,...,7
(1) x + y is a multiple of 9.
multiple of 9 = 9,18,27
suppose x+y = 18
x=15,y=3 then m=6, n=3 hence m+n = 9
suppose x+y = 27
x=14, y = 13 then m=5,n = 4 m+n =9
suppose x=15, n = 12; m=6,n=3 m+n = 9
hence 1 is enough then the answer could be A or D
(2) x*y divided by 9, the remainder is 5
x*y= (9p+m)*(9q+n)=81pq+9pn+9qn+mn from this we know that m*n is 5
so m must be 5 and n must be 1 only then m+n =6 enough
answer is D
