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I'm a new DM. One of the PCs obtained gauntlets of ogre power, which raise the Strength score of the creature wearing and attuned to them to 19. Does this increase the modifier or any of the skill checks or damage rolls as well?

I'm not sure how having higher Strength is helpful if it doesn't.

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    \$\begingroup\$ Welcome to RPG.SE! Take the tour if you haven't already, and check out the help center for more guidance. \$\endgroup\$ – V2Blast Sep 15 at 22:59
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The modifier is directly linked to the stat, so a STR of 19 has a modifier of +4. That just happens automatically. STR based attacks will therefore benefit from the higher modifier. Skill checks will improve if they use the STR modifier as it has improved.

Basically, the modifier instantly increases to match the new total of the stat then that flows over into attack rolls, damage rolls and ability checks/saves that use that modifier.

From the SRD:

When a character makes an attack roll, the two most common modifiers to the roll are an ability modifier and the character's proficiency bonus. When a monster makes an attack roll, it uses whatever modifier is provided in its stat block.

Ability Modifier. The ability modifier used for a melee weapon attack is Strength

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    \$\begingroup\$ Thank you. Your answer got me to look for a table in the PHB, and there is one (pg 173)! It shows modifiers based on ability scores. Our group used the premade PCs in the starter set, so having not made a character from scratch, was not sure how this worked. Thanks! – \$\endgroup\$ – Steven Sep 15 at 23:08
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    \$\begingroup\$ It will also increase things like carrying capacity and jump distance. \$\endgroup\$ – Nacht - Reinstate Monica Sep 16 at 12:24
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    \$\begingroup\$ @Steven Note that you don't need a table, the modifier is just (Score - 10)/2 rounded down. \$\endgroup\$ – Cubic Sep 16 at 15:19

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