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JEE Main 202621 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
21 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Permutation and Combination

JEE Main 2026 · 21 January · Morning Shift · Mathematics · Permutation and Combination

Let S=\ (m, n): m, n ∈\ 1,2,3, .., 50\ \ . If the number of...

Let S={(m,n):m,n∈{1,2,3,…..,50}}\mathrm{S}=\{(\mathrm{m}, \mathrm{n}): \mathrm{m}, \mathrm{n} \in\{1,2,3, \ldots.., 50\}\}. If the number of elements (m,n)(\mathrm{m}, \mathrm{n}) in S such that 6m+9n6^{\mathrm{m}}+9^{\mathrm{n}} is a multiple of 5 is pp and the number of elements (m,nm, n) in SS such that m+nm+n is a square of a prime number is q, then p+q\mathrm{p}+\mathrm{q} is equal to ____\_\_\_\_.

Correct answer

0

Step-by-step solution

For 6m+9n≡0(mod5)6^m + 9^n \equiv 0 \pmod{5}:
6≡1(mod5)6 \equiv 1 \pmod{5} and 9≡−1(mod5)9 \equiv -1 \pmod{5}
So 6m+9n≡1+(−1)n(mod5)6^m + 9^n \equiv 1 + (-1)^n \pmod{5}
This equals 0 only when n is odd.
Number of odd n in {1,...,50} = 25, and m can be any of 50 values.
p=50×25=1250p = 50 \times 25 = 1250
For m+nm + n = square of prime, possible values are 4, 9, 25, 49.
m+n=4m + n = 4: 3 pairs
m+n=9m + n = 9: 8 pairs
m+n=25m + n = 25: 24 pairs
m+n=49m + n = 49: 48 pairs
q=3+8+24+48=83q = 3 + 8 + 24 + 48 = 83
p+q=1250+83=1333p + q = 1250 + 83 = 1333
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