The sum of all two-digit positive numbers whi | vedclass

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(D) Two-digit numbers of the form $7n + 2$ are $16, 23, \dots, 93$. This is an arithmetic progression with first term $a = 16$, last term $l = 93$, an

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$1356$

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(D) Two-digit numbers of the form $7n + 2$ are $16, 23, \dots, 93$. This is an arithmetic progression with first term $a = 16$, last term $l = 93$, and common difference $d = 7$. The number of terms $n_1$ is given by $93 = 16 + (n_1 - 1)7$, which gives $77 = (n_1 - 1)7$, so $n_1 = 12$. The sum $S_1 = \frac{12}{2}(16 + 93) = 6 	imes 109 = 654$.Two-digit numbers of the form $7n + 5$ are $12, 19, \dots, 96$. This is an arithmetic progression with first term $a = 12$, last term $l = 96$, and common difference $d = 7$. The number of terms $n_2$ is given by $96 = 12 + (n_2 - 1)7$, which gives $84 = (n_2 - 1)7$, so $n_2 = 13$. The sum $S_2 = \frac{13}{2}(12 + 96) = \frac{13}{2} 	imes 108 = 13 	imes 54 = 702$.The total sum is $S_1 + S_2 = 654 + 702 = 1356$.

The sum of all two-digit positive numbers which, when divided by $7$, yield $2$ or $5$ as a remainder is:

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